Barnaby Joyce insists he and Malcolm Turnbull continue to have a strong working relationship despite a public spat over the deputy prime minister’s affair with an ex-staffer who is now pregnant with their child.

The Nationals leader told Fairfax Media “Malcolm and I are strong personalities” who can retain their working relationship after the pair met in private on Saturday to settle their differences.

“Like most people in a strong business relationship, there are times you need to discuss your views because that’s how you resolve things,” Mr Joyce said in the interview published on Monday.

“I don’t believe either of us are the sort of people who whisper behind closed doors – if you have an issue, you confront it head on and that’s what we did.”

A Newspoll published by The Australian on Monday found 65 per cent Australian voters believe Mr Joyce should quit as Nationals leader. The poll of 1632 voters was conducted between February 15 and 18.

Mr Joyce rejected ongoing questions about his travel expense and other entitlements.

“I am and continue to be confident that there has been no misuse of travel or entitlements, nor that any has been or will be found. I base that confidence on the fact that hundreds of inquiries have been made and nothing has been found,” he said.

Mr Joyce addressed questions surrounding the presence of his now partner Vikki Campion on the Sunshine Coast in January 2017 ahead of his appearance on the ABC’s 7.30 program and other media engagements, during a family holiday.

“It’s unsurprising that a media adviser would come to help you with media on 7.30 on and a major bio-security outbreak such as white spot,” he told Fairfax.

Mr Joyce on Monday began five days personal leave after a tumultuous week that culminated in a public falling out with Mr Turnbull and murmurings about his position as Nationals leader.

On Thursday, while announcing ministers would be banned from having sex with staffers, Mr Turnbull described Mr Joyce’s affair with Ms Campion as a “shocking error of judgment” that caused “a world of woe” for Mr Joyce’s wife and four daughters.

That provoked an angry response from Mr Joyce, who said the remarks were “inept” and “unnecessary”.

The government announced last week Mr Joyce would take personal leave and not stand in as acting prime minister when Mr Turnbull heads to the US later this week.

Deputy Liberal leader Julie Bishop acknowledged the past 10 days had been very distracting for the government.

But she said Mr Joyce’s future was in the hands of his Nationals colleagues.

Asked about the Newspoll result, she said: “I’m sure [Nationals politicians will] take a whole range of matters into account as they consider this issue”.

“… we’ll see how things turn out in the following days and weeks,” she told Sky News from London.

With Ms Bishop also away, Finance Minister Mathias Cormann will be acting prime minister.

– AAP