By Kathryn Snowdon, June 26: Matt Hancock's breach of social distancing guidance when he kissed an aide could damage government messaging on fighting the virus, families of Covid victims have warned.
The health secretary has apologised after pictures emerged of him with Gina Coladangelo, reportedly taken on 6 May.
The Covid-19 Bereaved Families for Justice group said failing to sack Mr Hancock could lead to a "Cummings effect" and people breaking the rules.
The PM considers the matter closed.
A Downing Street spokesman said Boris Johnson accepted Mr Hancock's apology, adding the prime minister had full confidence in the health secretary.
However, some Conservative MPs have expressed anger over the revelations and one former minister said they did not expect the health secretary to survive the scandal, the BBC's political correspondent Jonathan Blake said.
'Why would anyone listen?'
The Covid-19 Bereaved Families for Justice group has written to Mr Johnson urging him to sack Mr Hancock if he does not resign and has questioned whether the health secretary can now hold any moral authority in relation to Covid.
Rivka Gottlieb, from the campaign group, told BBC Radio 4's The World Tonight: "If he were to announce another lockdown or further regulations why would anybody listen to someone who doesn't follow the rules themselves? It's a bit like the Cummings effect."
Last March, the prime minister's then-most senior adviser, Dominic Cummings, drove 260 miles from his home in London to Durham during lockdown after he and his wife caught Covid - at a time when there were strict limits on travel.
Despite widespread condemnation, the PM stood by his key adviser, saying Mr Cummings had "no alternative" but to travel.
Ms Gottlieb said people had been "outraged" by Mr Cummings' actions, and she feared people would be more likely to break the rules if Mr Hancock remained in the post.
Social distancing at work is not a legal requirement, but the government recommends that people keep 2m apart where possible - or 1m with "risk mitigation", such as standing side by side or wearing masks.
Prof Stephen Reicher, from the University of St Andrews' School of Psychology, said what made the Cummings scenario "truly toxic" was not the government adviser's actions, but rather Mr Johnson's defence of them.
"An individual indiscretion turned into a systemic issue, a sense of there's one rule for us and another rule for them," Prof Reicher told BBC Two's Newsnight.
"In the midst of a huge national crisis, we want a government we trust and we want a government that we're going to listen to."
Cabinet colleagues of Hancock have defended the health secretary.
Housing Secretary Robert Jenrick told BBC Radio's 4 Any Questions programme: "There's a task to be done, Matt is on the job doing that, and I think we should allow him to get on with the job."
He added: "The rules have been hard. It is everybody's duty to follow the rules, but equally, I've not been somebody who has criticised and condemned people when they've made mistakes."
International Development Secretary Liz Truss told the BBC: "[Mr Hancock] does have my support."
Transport Secretary Grant Shapps said there was "a complete difference between what people do in their job... and what they do in their personal lives".
But Conservative commentator Tim Montgomerie, a former adviser to Mr Johnson, told BBC Radio 4's Today programme he believed Mr Hancock had been "a good secretary of state" but should resign.
"When you undermine your own rules you have to show to the public that you understand the transgression you've made and you resign," he said.
"He could come back in 12 to 18 months but if the public doesn't see some act of contrition, some sense that when rules are broken there are consequences, then confidence in the government, the Covid rules and the Conservative party are diminished."
Dave Penman, general secretary of the FDA union, which represents senior civil servants, told Today the relationship should have been disclosed.
Ms Coladangelo sits as a non-executive director of the Department of Health and "if that information is kept secret from them that really is unforgivable", he said.
"Inevitably, there would have been issues where there was the potential to challenge the secretary of state from the board. If one of those members was having an intimate relationship with the secretary of state, it completely undermines those decisions," he added.
Labour said Mr Hancock's position had become "hopelessly untenable" and has called for him to be sacked.
A Labour spokesman said the matter was "definitely not closed, despite the government's attempts to cover it up".
In May last year, epidemiologist Prof Neil Ferguson resigned from the government's scientific advisory group (Sage) after it emerged he had broken lockdown rules when a woman he was reportedly in a relationship with visited his home.
At the time, Mr Hancock said: "Professor Ferguson is a very, very eminent and impressive scientist and the science that he's done has been an important part of what we've listened to and I think that he took the right decision... to resign."
When asked by Sky News whether he thought Prof Ferguson should be kept in his position, Mr Hancock replied: "That's just not possible in this circumstance."
The Sun, which first published the photos of Mr Hancock and Ms Coladangelo, taken inside the Department of Health, carries further images of the pair on its front page on Saturday.
The paper says the latest image is of Mr Hancock and Ms Coladangelo at a restaurant on 23 May - 17 days after the picture of the kiss.
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