Trump promised to clarify what he called a "very simple" story after his newly recruited lawyer, Rudy Giuliani, revealed the president had reimbursed his personal attorney for the payment to Stormy Daniels, whose real name is Stephanie Clifford.
Trump denied last month knowing about the $130,000 payment, made on the eve the hotly contested 2016 US presidential election to silence Daniel's claims of a decade-old affair with Trump.
Instead he suggested Giuliani misspoke -- though in tweets on Thursday, Trump himself confirmed the payment and reimbursement to his long-time personal lawyer Michael Cohen.
"Virtually everything said has been said incorrectly and it's been said wrong or it's been covered wrong by the press," Trump said.
"Rudy's great, but Rudy had just started and he wasn't totally familiar with everything," Trump told reporters.
"He started yesterday. He'll get his facts straight."
'We're not changing stories'
Asked if he was changing his claim that he didn't know about the payments, Trump snapped back: "We're not changing any stories. All I'm telling you is that this country is right now running so smooth, and to be bringing up that kind of crap, and to be bringing up witch hunts all the time, that's all you want to talk about."
He said there "has been a lot of misinformation," but gave no specifics.
Giuliani was recruited to Trump's legal team two weeks ago to deal with special prosecutor Robert Mueller's push to interview the president in the investigation into possible Trump campaign collusion with Russia in the election, and obstruction allegations against Trump.
On Friday, Trump said he was willing to be interviewed by Mueller, but only if he gets fair treatment.
"I have to find that we're going to be treated fairly. Because everybody sees it now and it's a pure witch hunt."
"There was no collusion with the Russians, there was nothing, there was no obstruction," he added.
The Stormy Daniels issue has until now been a sideshow, with Cohen, for years a "fixer" at Trump's real estate group, bogged down in civil suits from Daniels and coming under a separate FBI probe for still unrevealed reasons.
Cohen last year said he had paid Daniels out of his own money in return for signing a non-disclosure agreement about her alleged 2006 affair with Trump.
But now questions have been raised over whether the payment violated election laws.
On Wednesday, Giuliani said on Fox News that Trump had reimbursed Cohen over a period of several months after assuming the presidency. He said the reimbursements came out of Trump's personal family accounts and so had not violated any campaign finance laws.
Trump then confirmed what Giuliani said via Twitter.
"Mr. Cohen, an attorney, received a monthly retainer, not from the campaign and having nothing to do with the campaign, from which he entered into, through reimbursement, a private contract between two parties, known as a non-disclosure agreement," he said.
"The agreement was used to stop the false and extortionist accusations made by her about an affair," he said.
Yet some analysts say the payment, made in the heat of the election, could nevertheless have violated laws because it sought to influence the outcome of the vote by keeping Stormy Daniels silent.
Public interest group Common Cause filed formal complaints in January calling for the Justice Department and Federal Election Commission to investigate the payment.
Giuliani "put President Trump in legal peril for 'knowing and willful' violations of campaign finance law related to hush money payments," said Paul Ryan, Common Cause vice president for litigation.
"Americans have a right to know who's spending money to influence their votes on Election Day. Team Trump's failure to disclosure this payment violated this right and the law."
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