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Hancock, who is married to wife Martha, has been accused of having an affair with aide Gina Coladangelo, who is also married to businessman Oliver Tress.
Speaking out for the first time since pictures emerged that appeared to show the pair kissing while at work, Hancock apologised for breaking Covid-19 guidelines.
He said: "I accept that I breached the social distancing guidance in these circumstances. I have let people down and am very sorry."
However, he also seemed to suggest that he will not be resigning after the incident, as well as the backlash that he has faced.
Hancock continued: "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."
A spokesperson for Boris Johnson said he 'considers the matter closed'.
"The Health Secretary set out that he accepted he had breached the social distancing guidelines and he has apologised for that," the spokesperson said:
"The Prime Minister has accepted the Health Secretary's apology and considers the matter closed."
Asked whether Boris Johnson had 'full confidence' in Hancock, the spokesman replied: "Yes."
Coladangelo and Hancock were allegedly seen in an intimate moment, in photographs obtained by The Sun through a whistleblower.
It claims that the pictures were taken in the office of the Department of Health in May of this year.
Ms Coladangelo has reportedly known Hancock since they were at university together, and was made a non-executive director at the Department of Health.
That roles comes with a salary of around £15,000 for between 15 and 20 days of work each year.
She is also the marketing and communications director of homeware and fashion chain Oliver Bonas, which was founded by her husband, Oliver Tress.
A government spokesperson confirmed that her appointment to the Department of Health was 'made in the usual way' and 'followed correct procedure'.
Anneliese Dodds, the chairwoman of the Labour Party, led calls for Hancock to resign after the scandal emerged.
She said: "If Matt Hancock has been secretly having a relationship with an adviser in his office - whom 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."
Liberal Democrat Health spokesperson Munira Wilson also said: "He was telling families not to hug loved ones, while doing whatever he liked in the workplace."
Matt Hancock has been married to wife Martha, who works as an osteopath, for 15 years.
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