After photos of Gina Coladangelo and him were reportedly taken on May 6, the health secretary apologized.
Duncan Baker is the Tory's first MP to tell the government openly that he should resign. Labour and the Covid-19 Bereaved Families for Justice group also called for his resignation.
The PM declares the matter closed.
A Downing Street spokesperson said that Boris Johnson had accepted Mr Hancock’s apology and added that the prime minister had full faith in the health secretary.
The Sun published photos of Mr Hancock kissing Ms Coladangelo on Friday. They are both married and have three children. They were taken by the Department of Health on May 6, according to the Sun. The health secretary said he was sorry.
North Norfolk's MP, Mr Baker, told his local newspaper, the Eastern Daily Press he felt the health secretary had "failed" to implement a few measures.
He stated, "I will not condone such behavior and I have in all the strongest terms told the government my thoughts."
Former adviser to Johnson, Tim Montgomerie, Conservative commentator, said that Mr Hancock was "a good secretary-of-state" and should resign.
He said, "When you violate your own rules you must show the public you understand the transgression and you resign."
Later, Mr Montgomerie told BBC News that Mr Hancock would likely survive as a result of concerns at No10 about his sacking leading to increased scrutiny around other ministers. This could create a domino effect.
The families of Covid victims have warned that the violation of social-distancing guidelines could cause damage to government messages on the virus.
Rivka Gottlieb, of the Covid-19 Bereaved Families for Justice Group, said on BBC Radio 4's The World Tonight that if he announced another lockdown or additional regulations, why would anyone listen to someone who doesn’t follow the rules?
She compared it with Dominic Cummings, the former prime minister's senior adviser, who traveled from London to Durham in lockdown last March. Despite widespread condemnation of his actions, Johnson remained loyal to his aide.
Cabinet colleagues have supported the health secretary, with Robert Jenrick, Liz Truss, International Development Secretary Liz Truss, and Grant Shapps backing him publicly.
There have been concerns about the process by which Ms Coladangelo was appointed as a non executive director of Department of Health.
Ms. Coladangelo, a friend of the secretary of health since they worked together on a student radio station at Oxford University, was appointed to the position last September. It comes with a PS15,000 salary. She also has 15 to 20 working days per year.
Angela Rayner, Labour's deputy leader, has written to the cabinet secretary (the UK's highest ranking civil servant) asking him to investigate whether Mr Hancock violated ministerial rules by failing declare the relationship.
The spokesperson for No. 10 insisted that the correct procedure had been followed, but refused to give any details.
Today was told by Dave Penman, the general secretary of the FDA union which represents senior civil servants. He said that the relationship should have been made public, but that if the information is kept from them, it is "unforgiveable".
"Inevitably there would have been problems where there was the possibility to challenge the secretary-state from the board. He said that if one of the members had an intimate relationship to the secretary of states, it would completely undermine those decisions."
Labour claimed that Mr Hancock's situation was "hopelessly untenable" so he should be fired.
Kirsten Oswald, the SNP's Westminster deputy leader, said that the prime minister "risks jeopardising vital government health measures" by keeping Mr Hancock as Health Secretary. She stated that there must be public trust in those who set the rules.
Liberal Democrats stated that Mr Hancock should have been fired for "his failures" years ago and added, "This latest episode of hypocrisy will destroy the trust with British citizens".
