Matt Hancock has apologised for breaking social distancing rules after he was pictured kissing a close aide in his Whitehall department.
However, the Health Secretary made clear he intends to resist calls for his resignation following reports he was having an extramarital affair with Gina Coladangelo, who he appointed last year.
In a brief statement, Mr Hancock said he is “very sorry” that he has let people down, but that he remains focused on his job tackling the pandemic.
“I accept that I breached the social distancing guidance in these circumstances, I have let people down and am very sorry,” he said.
“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.”
But Labour said his position has become “hopelessly untenable” and called on Boris Johnson to sack him.
The Sun published images, apparently captured from CCTV footage, of the couple together which are said to have been taken on May 6 at the headquarters of the Department of Health and Social Care.
The Health Secretary hired Ms Coladangelo as an unpaid adviser on a six-month contract in March last year, before appointing her as a non-executive director at the department.
Mr Hancock, who met Ms Coladangelo at Oxford University where they both worked on the student radio station, has been married to his wife Martha for 15 years and they have three children.
Ms Coladangelo is the marketing and communications director at Oliver Bonas, a British retailer founded by her husband, Oliver Tress.
Labour Party chair Anneliese Dodds said in a statement: “If Matt Hancock has been secretly having a relationship with an adviser in his office – who he personally appointed to a taxpayer-funded role – it is a blatant abuse of power and a clear conflict of interest.
“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.”
Following Mr Hancock’s statement, she added: “He set the rules. He admits he broke them. He has to go. If he won’t resign, the PM should sack him.”
Liberal Democrat leader Sir Ed Davey said on Twitter: “The reason Matt Hancock should resign is that he is a terrible Health Secretary, not because of his private life.
“From the PPE scandal, the crisis in our care service and the unbelievably poor Test and Trace system, he has utterly failed.”
Earlier, Transport Secretary Grant Shapps said reports about his Cabinet colleague are a personal matter as he insisted Ms Coladangelo would have gone through an “incredibly rigorous process” to get the role.
“First of all, I think the actual issue is entirely personal for Matt Hancock,” he told Sky News.
“In terms of rules, anyone who has been appointed has to go through an incredibly rigorous process in Government, so, whatever the rules are, the rules will have to be followed.
“There are no short cuts to that, as anyone who has had anything to do with the appointments system in the Civil Service knows.”
He said there are “very strict rules in place” in terms of how advisers are appointed, adding: “I think it is a bit of red herring in this case.”
It emerged in May last year that Government adviser Professor Neil Ferguson had allowed a woman, reported to be his lover, to visit him at home in London on at least two occasions during lockdown.
The Government adviser, whose work was crucial in Boris Johnson’s move to enforce strict restrictions, stood down from the Scientific Advisory Group for Emergencies (Sage) over what he called an “error of judgment”.
Mr Hancock called the revelations “extraordinary” at the time, telling Sky News: “Everyone has got to follow the social distancing rules.”