I have had a good look at The Times article about the emails. I can't link it because it's behind a paywall - I've tried copying but have only managed bits (so it's not the full story).
They knew last December that MH was using personal accounts to run Department business -warned him, noted that they would be able to properly track negotiations/communications and gave him another account to use but he carried on using personal accounts whilst understanding that it would look shady.
I think MH is guilty of a lot more than an pathetic office affair.
Matt Hancock faces an investigation after using a personal email account instead of an official address during the pandemic in a breach of government guidelines.
Since March last year the former health secretary has routinely used a private account to conduct government business, concealing information from his own officials and potentially the public, according to documents obtained by The Sunday Times.
It means that the government does not hold records of much of Hancock’s decision-making, including negotiating multimillion-pound PPE contracts, setting up the £37 billion test and trace programme and overseeing the government’s care homes strategy.
The minutes concern a meeting about a Good Law Project legal challenge over Hancock’s decision to award a contract worth up to £75 million for “malfunctioning” tests to a firm linked to Sir John Bell, a government adviser.
In the minutes, Williams admits that he “doesn’t believe there was inappropriate acts on behalf of ministers but can clearly see the optics suggest otherwise”.
Since the meeting, Hancock has been given an official email account, although two Whitehall sources said that he still preferred to use Gmail. This is considered to be a less traceable form of communication.
Under the Freedom of Information Act, officials can request access to private email accounts. Yet the Information Commissioners Office (ICO) says such searches are rare. The individual must be asked to go through their own personal address and decide what to disclose, rather than handing it over to officials in full
The disclosures pose new questions about Hancock’s conduct. They suggest it will be difficult for officials to obtain evidence of his conduct in office before an independent inquiry into the government’s handling of the coronavirus pandemic.
His emails may also be important for any investigation into whether he breached MPs’ rules by sponsoring a parliamentary pass for Colandangelo. Cabinet Office guidance states explicitly that “it is expected that government business should be recorded on a government record system”.