The photos, published by the Sun, showed the health secretary in a clinch with Gina Coladangelo on May 6, when people were still not allowed to meet or hug indoors.
Labour and the Liberal Democrats said Hancock’s position was now untenable and urged Boris Johnson to sack him.
But the health secretary looked set to dig in and try and hang on to his job.
He said: “I accept that I breached the social distancing guidance in these circumstances.
“I have let people down and am very sorry.
“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.”
Hancock was pictured kissing Coladangelo on May 6.
It was not until May 17 that rules were relaxed to allow friends and family to meet indoors, and to enjoy close personal contact like hugging.
Labour said Hancock had abused his power and broken Covid rules.
Party chair Anneliese Dodds said: “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.”
Lib Dem health spokesperson Munira Wilson said: “Matt Hancock is a terrible health secretary and should have been sacked a long time ago for his failures.
“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.”
According to the Sunday Times, Coladangelo was hired by Hancock in March 2020 as an adviser before being appointed in September as a non-executive director role, which will see her earn at least £15,000 of taxpayers’ money, which could rise by a further £5,000.
Coladangelo is also marketing and communications director at retailer Oliver Bonas, and is reportedly a major shareholder in lobbying firm Luther Pendragon.
Hancock has already been under pressure following a string of attacks from Johnson’s former adviser Dominic Cummings.
Cummings accused Hancock of repeatedly lying to ministers and officials during the Covid pandemic, which the health secretary denies, and recently published WhatsApp messages which revealed that the prime minister thought his Cabinet colleague’s performance was “totally fucking hopeless”.