It was revealed by The Sun that Mr Hancock had been conducting an extra-marital affair with Gina Coladangelo during the Covid pandemic.
Ms Coladangelo is communications director at lifestyle company Oliver Bonas, which was founded by her husband Oliver Tress.
It was also claimed she was a director and major shareholder at lobbying firm Luther Pendragon, however the firm has since revealed she resigned as a director in 2017.
Mr Hancock, who met Ms Coladangelo at Oxford University in the early 2000s, is accused of “secretly appointing" her to the Department of Health and Social Care as an unpaid adviser on a six-month contract in March last year. In her role she accompanied him to confidential meetings with civil servants and visited No.10.
He then made her a non-executive director at the department in September, though there is no public record of the appointment.
The SNP is now demanding the UK Government launch an investigation into how Ms Coladangelo was given the series of roles at the heart of government, including a £15,000 role on the board.
The party has also suggested Mr Hancock broke the ministerial code which states that “on appointment to each new office, Ministers must provide their Permanent Secretary with a full list in writing of all interests which might be thought to give rise to a conflict.”The list should also cover interests of the Minister’s spouse or partner and close family which might be thought to give rise to a conflict.”
However Mr Hancock’s Cabinet colleague, Grant Schapps, said she would have passed a “rigorous” recruitment process and Boris Johnson has reportedly accepted an apology from Mr Hancock and believes the matter to be closed.
Edinburgh East MP Tommy Sheppard said the public deserved transparency as to why “so many Tory friends and donors have been given jobs, peerages, public contracts and millions – if not billions – of pounds in taxpayers' money”.
Mr Sheppard said: “Private matters are just that, but public appointments are another matter entirely – and they warrant proper scrutiny and full transparency.
"There must be an investigation into this appointment and a full public inquiry into the Tory cronyism scandal engulfing Westminster, which is out of control. The public deserve answers.
"It might be an amazing coincidence that the Tory health secretary's university friend was the best person for a £15,000 role on the board of his department and also as an adviser – but it raises more questions over why Tory friends and donors are being given privileged access and public money.”
He added: “Whether it's unlawfully handing multi-million pound contracts to friends, soliciting donations to redecorate the Downing Street flat, offering peerages to billionaire donors, organising tax breaks over text, or giving your pals jobs at the heart of the UK Government – many will conclude this Tory chumocracy shows Boris Johnson and his ministers are arrogantly abusing public office and are only in it for themselves.”
The Sun published pictures of the married Cabinet minister appearing to kiss Ms Coladangelo. The images, which appear to be captured from CCTV footage, were taken on May 6 from the headquarters of the Department of Health and Social Care.
It is the second cronyism row in two months for Mr Hancock. He was previously accused of not registering share ownership in his sister’s company, which won a place on a government framework to provide services to the English NHS in 2019. The claim was rejected by the UK Government.
Last week it was also revealed by Dominic Cummings, the former adviser to Boris Johnson, that the Prime Minister had called the health secretary “hopeless” over his handling of the pandemic. The PM's official spokesman later insisted Mr Johnson had full confidence in Mr Hancock.
Labour said the government needed to answer whether the health secretary had broken any rules or there had been "conflicts of interest" in the appointment of his closest adviser.
A party spokesman 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.”
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.”
Mr Hancock was not at his north London home on Friday morning. The DHSC has been asked for comment.
A statement from the lobbying firm Luther Pendrago said that Ms Coladangelo had ceased to be an employee in 2014 and resigned as a director in 2017.
"She has no involvement in the day to day running of Luther Pendragon, nor does she carry out any client work on behalf of Luther Pendragon.
"She receives no remuneration from Luther Pendragon, and has not done so since she ceased being an employee.
“She is a minor, not a major, shareholder in Luther Pendragon.”
Mr Hancock has also come in for criticism over the awarding of government contracts to suppliers of PPE.
He was unapologetic after the High Court ruled the UK Government unlawfully failed to publish details of billions of pounds' worth of coronavirus-related contracts.
Asked if he had anything to apologise for despite losing the case, Mr Hancock told the BBC: "People can make up their own view about whether I should have told my team to stop buying PPE and spend the time bringing forward those transparency returns by just over a fortnight.
"Or whether I was right to buy the PPE and get it to the front line, you tell me that that is wrong. You can't. And the reason you can't is because it was the right thing to do.”