UK Health Secretary Matt Hancock apologised for breaking social distancing rules after pictures showed him hugging and kissing an aide.
The Sun published CCTV photos of Hancock embracing Gina Coladangelo, allegedly in his Whitehall office last month.
The pair – friends from the University of Oxford – are married to and have children with other people.
Mr Hancock, who is facing calls for his resignation, apologised for breaking social distancing rules.
"I accept that I breached the social distancing guidance in these circumstances,” he said.
"I have let people down and am very sorry."
He said he had no intention of stepping down.
“I remain focused on working to get the country out of this pandemic, and would be grateful for privacy for my family on this personal matter,” he said.
Downing Street said Prime Minister Boris Johnson accepted Mr Hancock's apology and "considers the matter closed".
Asked whether the prime minister had "full confidence" in Mr Hancock, Mr Johnson's spokesman said: "Yes."
The main opposition Labour Party, which has accused the government of cronyism in awarding millions of pounds of contracts related to the pandemic, said Mr Hancock's position was untenable.
"If Matt Hancock has been secretly having a relationship with an adviser in his office – whom he personally appointed to a taxpayer-funded role – it is a blatant abuse of power and a clear conflict of interest,” said Labour’s chairwoman Annaliese Dodds.
"The charge sheet against Matt Hancock includes wasting taxpayers' money, leaving care homes exposed and now being accused of breaking his own Covid rules.
"His position is hopelessly untenable. Boris Johnson should sack him."
Earlier, Transport Secretary Grant Shapps described it asa personal matter.
“What somebody does in their private life is their private business,” he told Times Radio, during Friday’s ministerial media round.
“We’re living in the 21st century.”
Asked on LBC radio whether the health secretary had been "ignoring social distancing" with Ms Coladangelo, Mr Shapps said: "I'm quite sure that whatever the rules were at the time were followed.
"You'll recall that there was a point at which social distancing rules were changed but, as I say, I don't want to comment on somebody else's private life – that is for them.”
Ms Coladangelo, a former director at lobbying agency Luther Pendragon and current shareholder, was appointed by Mr Hancock as an unpaid adviser to the Department of Health last year.
She was later made a non-executive director at the department, a role that pays £15,000 ($21,000) a year, reports suggested. Mr Hancock chairs the departmental board.
"Ministers, like everyone, are entitled to a private life," a Labour spokesman said.
"However, when taxpayers' money is involved or jobs are being offered to close friends who are in a personal relationship with a minister, then that needs to be looked into."
The Sun did not say how it obtained the security camera images, but it cited a whistle-blower as commenting on the relationship.
The revelations will pile pressure on Mr Hancock at a delicate time after repeated criticisms of his handling of the coronavirus pandemic.
It emerged this month that Mr Johnson in March branded Mr Hancock "hopeless" in a private message to Dominic Cummings, the prime minister's adviser at the time.
Mr Hancock defended his response to the coronavirus pandemic in a parliamentary hearing this month, saying he had acted with "honesty and integrity" throughout the crisis.
The health secretary was also accused of cronyism in April after NHS Wales awarded two contracts to a company in which he owns shares. His sister is a director of the company.