Mr Hancock is in the spotlight after being caught on camera embracing Gina Coladangelo, a university friend he appointed to a director’s role at the health department.
The health secretary was already facing questions about his appointment of Ms Coladangelo to the £15,000-a-year role, without it being properly announced last year.
Mr Hancock – who has been married for 15 years to Martha and has three children with her – has yet to comment on the allegation of an office affair.
Grant Shapps, the transport secretary, argued it was an “entirely personal issue for Matt Hancock” and insisted all appointments went through “an incredibly rigorous process”.
“The health secretary been working very hard rolling out this vaccine programme – and I’ll leave it there,” he told Sky News.
But a Labour spokesperson said: “Ministers, like everyone, are entitled to a private life.
“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 government needs to be open and transparent about whether there are any conflicts of interests or rules that have been broken.”
At the time the photo was taken, indoor gatherings between people from different households were against the law, unless they came under an exemption such as being “reasonably necessary” for work.
Advice against hugging loved ones in England was not lifted until 11 days later on 17 May.
Official guidance said that people who need to meet at work should remain two metres apart - or one metre if mitigating measures such as face-coverings were used.
Liberal Democrats called for Mr Hancock to go.
“Matt Hancock is a terrible health secretary and should have been sacked a long time ago for his failures,” said Lib Dem health spokesperson Munira Wilson.
“This latest episode of hypocrisy will break the trust with the British public. He was telling families not to hug loved ones, while doing whatever he liked in the workplace.
“It’s clear that he does not share the public’s values. Rules for them and rules for us is no way to run a country.
“From the PPE scandal, the crisis in our care service and the unbelievably poor test and trace system, he has utterly failed. It is time for the health secretary to go.”
Mr Hancock this morning cancelled a planned visit to a Covid vaccine centre at Newmarket Racecourse in Suffolk.
He described the professor’s behaviour as “extraordinary”, leaving him “speechless” and said: “I think he took the right decision to resign.”
Ms Coladangelo, who met Mr Hancock at Oxford University, was a director at lobbying firm Luther Pendragon, before he quietly made her an unpaid adviser at the Department of Health and Social Care, in March last year.
In September, he promoted her to being a non-executive director at the department, meaning that she is a member of the board that scrutinises the department.
There was controversy over the lack of any public record of the appointment, which gave Ms Coladangelo a parliamentary pass and unregulated access to the Palace of Westminster.
The government was already facing allegations of “chumocracy” and a lack of transparency in appointing friends from the private sector to key roles.
Lord Evans, the head of the committee on standards in public life, warned of a perception of disregarding “the norms of ethics and propriety that have explicitly governed public life for the last 25 years”.
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