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Boris Johnson has refused three times to deny outright he had an affair with Jennifer Arcuri - the US businesswoman being investigated for receiving public funds.

The prime minister repeatedly failed to deny having had a relationship with her while married, in an interview with Sky News.

Ms Arcuri was given a total of £126,000 in taxpayers' cash, privileged access to three overseas trade missions led by Mr Johnson and called him "one of her best friends", The Sunday Times revealed this weekend.

That sparked an investigation by the government's culture department into a £100,000 grant her company, Hacker House, won.

And the Independent Office for Police Conduct (IOPC) has been asked to decide if Mr Johnson should be investigated for the criminal offence of misconduct in public office.

Ms Arcuri's husband, Matthew Hickey, said on Twitter she had not received any money.

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But in comments that will likely cast a shadow over Mr Johnson's first speech since becoming prime minister at the Conservative Party conference in Manchester, the prime minister would not deny having had an affair with Ms Arcuri.

"I've said what I have to say about that matter," he told Sky News' political editor Beth Rigby.

"I think perhaps the most important point is that I'm very, very proud of everything we did in in London."

Jennifer Arcuri

Who is she and what is her connection to Boris Johnson?

Pressed to confirm he was not denying the affair, Mr Johnson said: "The crucial thing is that in terms of promoting London, everything was done with complete propriety."

Asked about the accusation from Labour he misused public funds, Mr Johnson said: "I can certainly say there was absolutely no question of that at all."

Mr Johnson also told Sky News he would propose new customs checks on the island of Ireland as part of any new Brexit deal.

He said that while he did not want to have "any kinds of checks at all at the border" there would have to be "checks at either end" as he confirmed he wanted to scrap the "backstop" deal agreed by his predecessor.

But the PM went on to insist that the checks would "not involve border posts away from the border" or infrastructure.

Mr Johnson did acknowledge that the principle of customs checks on the island of Ireland was a clear policy break from the May era but said the "tough bit" had to be done.

"This is the moment of truth, really, because in the end, the country has to be able to govern its customs," he said.

"If you're going to come out of the EU, you've got to run your own trade policy, you've got to run your own customs. So we have to find your own way of doing that. I won't deny it that this is the tough bit."

He confirmed that his new proposals will do away with the EU-UK Joint Report agreed in December 2017 to have no hard border or related checks or controls, preserving the all-island economy.

"The reality is, well, you can't make both things work at once. You have to accept that there's got to be a change," he said, adding it would require "a great deal of positive energy to solve".

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Mr Johnson admitted Brexit had proved particularly divisive in his family, after his brother Jo Johnson quit the government just weeks ago over the PM's plan.

"We're a very tight knit, close knit family," he told Sky News.

"And we all love each other very much. But there's no doubt about it - we do disagree about this issue. I mean, there are other members of my family who are Leavers but yes, in common with many families around the country there are difference of opinion."