In a blog post, Mr Cummings also alleged the prime minister called the track and trace system "like whistling in the dark" and worried about the UK gaining the "double distinction of being the European country w[ith] the most fatalities and the biggest economic hit."
The allegations will add extra pressure on the Health Secretary who has apologised for breaking social distancing guidance after The Sun newspaper released photos purportedly showing him having a ‘secret affair’ with his closest aide.
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Mr Cummings, who left Downing Street in November during a bitter power struggle in No 10, has targeted much of his criticism since leaving at the health secretary.
And in a hearing before the joint Science and Technology committee last month, Mr Cummings said he had told the prime minister to fire Mr Hancock on multiple occasions.
In the latest blog post, he claims he told the PM on May 7 2020 that "Hancock was ‘unfit for this job’ and him staying in place was ‘killing god knows how many’."
He added: "The PM agreed that Hancock’s failures were a catastrophe but refused to fire him."
Mr Cummings also included WhatsApp messages he claims he sent to a small group of advisors in Number 10, including the prime minister.
In one, sent on May 3 2020, he warned the group about his fears "we are negligently killing the most vulnerable who we are supposed to be shielding".
Boris Johnson replied to the message asking to see data from the ONS but did not address the point raised.
Mr Cummings also published a message he claimed was from Mr Johnson criticising the test and trace system.
In the message from April 26 2020, the prime minister is alleged to have written: "The whole track and trace thing feels like whistling in the dark"
"Legions of imaginary clouseaus and no plan to hire them.
"Apps that don’t yet work
"And above all no idea how to get new cases down to a manageable level or how long it will take"
"By which time uk may have secured double distinction of being the European country w[ith] the most fatalities and the biggest economic hit"
He purportedly added: "We GOTTA turn it round"
Mr Cummings criticised the "political institutions" in the UK and said they left the country to get "Hancocked".
He added: "Britain gets ‘Hancocked’ every week."
The Department of Health and Social Care did not immediately respond to a request for comment.