At his hectic 77-minute news conference last week, United States President Donald Trump momentarily confused his ally Australia with another country starting and ending with A.

Discussing the fact that details of two of his early phone calls with foreign leaders – from Mexico and Australia – had leaked, he described his reaction.

“When I was called out on Mexico, I was shocked,” Trump told reporters. “… But it wasn’t that important a call … Same thing with Australia; I said, ‘That’s terrible that it was leaked but it wasn’t that important.’ ”

He had ordered an investigation, fearful more important future conversations might also leak.

“So I’m dealing with Mexico,” the president continued. “I’m dealing with Argentina …”

The Australian officials listening to every word of his news conference knew what – and who – he meant. But it did serve as a reminder of this new president’s style: not being too bothered with the finer details – not publicly, anyway – and possibly requiring occasional prompting on who his country’s close friends are and why they matter.

In the case of Australia, one month since inauguration day that prompting is under way.

While the foundations of the Australia–US alliance hold from one government to the next, it’s the personal relationships that determine how smoothly things run and how much influence the smaller partner’s views are afforded in Washington. The dash across the Pacific by Foreign Minister Julie Bishop this week, for meetings with Vice-President Mike Pence and Secretary of State Rex Tillerson, was all about that.

“The vice-president and the administration more generally are asking for ideas from people,” Bishop told The Saturday Paper from Washington. “… I took the opportunity to share some of our thinking. They appeared to welcome it.”

It appears the leak of the contents of Trump’s phone call with Prime Minister Malcolm Turnbull is proving to be more help than hindrance, placing the Australia relationship temporarily front of mind among senior administration figures.

But there is a view among some in Washington that Turnbull erred in choosing to raise the refugee deal in the way he did in the first place.

While they maintain Trump’s reaction and the subsequent leak were unacceptable, they say it would have been wiser for Turnbull to have let officials press the case on the refugee deal and used his own contact with the new president to reinforce what Australia gave in the alliance relationship, not what it wanted to take.

They point to Japanese prime minister Shinzō Abe’s diplomacy as a better example, consulting regional leaders including Turnbull on issues of concern before jetting to Washington.

In that regard, the leak saved Turnbull. The widespread reporting of Trump’s intemperate response heralded a flood of goodwill and reassurance. Everywhere from TV talk show studios to the halls of the Capitol, Americans spoke out about Australia’s importance, and people asked privately and publicly: if we can’t get along with Australia, who can we get along with?

Theatre and substance

The nature of the Australian government’s approach to the new administration is encapsulated nicely in the title of a paper the Australian National University’s national security college published recently about the new presidency’s security challenges for Australia: “Don’t panic, don’t relax”.

Central to the government’s approach is the premise that despite predictions to the contrary, Trump’s behaviour is not going to change. As one official put it, it has worked for him in real estate, in reality television and in his personal life, so why would he change it? What’s more, he was elected declaring he was not a conventional politician and he can’t afford to become one.

Rather than dwelling on Trump’s rambunctious, unorthodox style and self-contradictions – and until the opportunity arises for Turnbull and Trump to meet face-to-face at an international summit or in Washington some time this year – Australian ministers and officials are focused on forming relationships with those around him.

They point out that Trump did not say anything during his combative news conference that impinged on Australia’s interests, and warn that observers “need to separate out theatre from substance”.

And they describe Trump’s key recent appointees as solid, praising the experience of both former general James Mattis and former oil executive Tillerson.

One official told The Saturday Paper the replacement of Mike Flynn as Trump’s national security adviser with Lieutenant-General Herbert Raymond “H.R” McMaster “could not have been better”. And officials are confident the phone call leak has guaranteed the refugee deal will be upheld and the US will, indeed, take up to 1250 people from the Nauru and Manus Island detention centres while Australia takes an unspecified number of refugees from what is known as the Central American “northern triangle” of El Salvador, Honduras and Guatemala, currently residing in camps in Costa Rica.

The phone call and its surrounding publicity also prompted a bipartisan resolution in the US congress, celebrating the two countries’ security alliance, and gave Australia a fleeting prominence in Washington that it would not otherwise have had so soon into this settling-in period.

Ministers have been quick to capitalise. Between meetings with senior members of the administration this week, including Pence and Tillerson, Julie Bishop said both sides had moved on from the phone call. “There’s a view in the administration that the reports have been exaggerated,” Bishop said.

Having had several less robust phone conversations of her own with both Pence and Tillerson, Bishop emphasised the importance of meeting both men in person. “It’s in our interests to have the closest possible connections,” she said.

Beazley concern

One month into the new administration, there has been no adjustment to Australian policy settings, no change to how intelligence is handled and, government sources insist, no shift in the political approach an Australian government always takes to forging ties with a new US administration.

But for all the nothing-to-see-here protestations, there is a laser-like focus on the bilateral relationship and The Saturday Paper has been told the government is not ruling out adjustments down the track.

Malcolm Turnbull is understood to be in constant contact with Australia’s ambassador in Washington, Joe Hockey.

Hockey’s predecessor, Kim Beazley, has lamented the lack of a substantive one-on-one conversation between Turnbull and Trump on matters beyond the refugee deal, which stole oxygen that should have been devoted to other issues.

“It won’t be easy or pleasant for Mr Turnbull to resume the conversation,” Beazley wrote last week in the Australian Strategic Policy Institute’s journal, The Strategist.

He said the outpouring of support and Turnbull’s low-key handling of the fallout would help. “The conversation, when a peg develops to hang it on, must resume.”

Iraq commitments

In the meantime, the vice-president is a particularly important connection, as someone who has the president’s ear.

The Australian side took it as a good sign that Pence, back from Europe only the night before, had squeezed in a meeting with Bishop and her officials in his White House office and let talks scheduled for half-an-hour run twice as long. It was, they said, a “very warm” engagement.

The notorious Trump–Turnbull phone call was not raised in Bishop’s meetings. Nor was the refugee deal, which she said was “proceeding” and being dealt with “at officials level”.

What was raised, and is being dealt with much higher up, is both the new US government’s attitude to engagement with Asia – especially in relation to North Korea and the South China Sea – and the pressing question of future military commitments in the Middle East.

“They are clearly reviewing all current policy positions in relation to their commitments overseas,” Bishop said. “I thought it was appropriate that we provide our thoughts.”

She declined to detail them but explained they related, in part, to how success against Daesh in Iraq would be defined and what would happen after.

It has become accepted wisdom regarding the previous Iraq conflict that too little postwar planning was done, with disastrous results that allowed Daesh to rise and flourish.

With the Iraqi city of Mosul on the brink of being seized back from Daesh, and Donald Trump talking up a rapid defeat of the group across the world, the Australian government is moving to ensure its existing commitments are acknowledged and its views heard.

Successive Australian governments have been reluctant to become involved in postwar reconstruction – what former prime minister John Howard used to call “nation-building”. Yet, in Afghanistan, that’s exactly what he did, in a commitment that saw Australian operational forces there for more than a decade.

In a possible sign of requests to come, the commander of international forces in Afghanistan, General John Nicholson, told the US senate armed services committee two weeks ago he needed “a few thousand” more personnel.

Specifically, he needed them to train and advise the Iraqi army – tasks Australians and others are doing there now – and said that perhaps allies could be asked to contribute.

Asked about Australia’s commitments, Bishop said: “I wasn’t asked to make any change.” While she was not expecting any request for more from Australia, she added: “Our military deployments are always under review.”

The US’s whole Middle East strategy is under much more active examination. The 30-day review President Trump ordered is due to conclude in about a week.

Like government officials, analysts in Australia are watching closely. Beyond the battle for Mosul, there are moves to also blast Daesh out of Raqqa in Syria.

One analyst told The Saturday Paper he would not be surprised if the US moved to put more forces on the ground in Syria.

James Mattis a steady hand

Defence Minister Marise Payne has been in talks with her US counterpart, too, meeting Mattis during NATO talks in Brussels last week.

Payne says she has not heard any suggestion of extra troops for Syria but that future operations are “very much part of the developments they are putting together in the report they are preparing for the president”.

“I’m not going to pre-empt that because we don’t have a presentation to the president yet which will form the basis for a discussion,” she said.

Like Julie Bishop, Payne made Australia’s views known. “Those views are being taken on board,” she said.

Mattis, a widely respected former general, has a very different personal style to that of his president. “He takes a very considered, very strategic approach to these things,” Payne said. “He knows it is very important to be very measured and careful in public commentary.”

Some observers note there is a kind of split personality to the administration so far, featuring on one side the steady, conservative approach of Pence, Mattis, Tillerson and White House chief of staff Reince Priebus, and on the other the more unpredictable former chief of hard-right website Breitbart and now senior presidential adviser Steve Bannon and Trump himself.

Mattis, Tillerson and Pence have all been in Europe in the past week calming fears of dramatic policy changes on engagement with NATO and the European Union or possible action in the South China Sea. Both Bishop and Payne pressed the need for de-escalation. Both say they are confident the US remains committed to strategic engagement in the Asia-Pacific region. Payne points to Mattis’s recent visit to Japan and South Korea as proof.

But some say Australia is not well enough prepared for the possibility of a US policy shift that could increase pressure on Australia’s ties with China.

ANU professor of strategic studies Hugh White says there should be adjustments in both Australia’s policy and approach. “The government seems very focused on pretending that nothing has changed,” White told The Saturday Paper. “It’s perhaps not surprising that Malcolm Turnbull would be cautious in his public commentary. What is surprising is he’s upped his rhetoric in support of Donald Trump.”

White also points to China’s increasing assertiveness. “We have no model for how Australia conducts its relationships in Asia and protects its interests in Asia when the US is not the dominant force in Asia … Australia’s political leaders have got to stand up and acknowledge we’ve got a problem.”

Former defence force chief Sir Angus Houston told the National Press Club this week that Australia must hold fast to the alliance or face a doubling of military  spending in self-defence.

Washington-based Australian Andrew Shearer, who was foreign policy adviser to prime minister Tony Abbott and is now with the Centre for Strategic and International Studies, says Australia must “work harder at the alliance”.

“We get enormous benefits from the alliance but we only get those benefits if we are prepared to be useful,” he told The Saturday Paper. He meant through diplomacy, intelligence and military contributions.

“There’s been a lot of inconsistent messages out of the Trump administration,” he said, “especially on alliances, but one thing that is clear is allies across the board are going to have to be prepared to lift their game.”

Security analyst Keith Suter suggested all was not quite as relaxed in the security community about the risks in Trump’s hot-headed public engagement – including his outbursts on Twitter – as the government’s soothing rhetoric would suggest.

“It’s clearly worrying that we’ve got this person in the White House who has difficulty sleeping,” Suter told Sky News. “And he’s giving the rest of us difficulty sleeping as well.”