The blog post exceeding 7,000 words is the latest salvo in a bitter row between the former aide, Mr Johnson and the health secretary, who Mr Cummings blames for many failings during the crisis.
The revelations came in a blog post by former Downing Street aide Dominic Cummings, which included a series of screenshots purportedly showing WhatsApp exchanges between himself and the Prime Minister as ministers grappled with the crisis in spring past year.
In one exchange on March 3rd a year ago, Mr Cummings highlighted the US's rapid increase of testing capacity and criticised Mr Hancock for saying he was "sceptical" about meeting a new United Kingdom target having earlier said it would "definitely" be met.
"I can't think of anything except taking Hancock off and putting Gove on", Mr Johnson apparently added in the conversation on April 27th a year ago.
Hancock last week denied he had ever said anything to Johnson he knew to be untrue.
The prime minister apparently responded, "Totally (obscenity) hopeless" to a comment by his ex-aide that Hancock was unsure he would reach a target of 10,000 virus tests per week on schedule.
In a second screenshot that Cummings shared on his blog, Johnson allegedly claimed the procurement process surrounding personal protective equipment (PPE) had been a "disaster".
When asked whether he was hopeless by reporters today, Hancock said "I don't think so".
Cummings left Downing Street past year after a behind-the-scenes power struggle with Johnson and his wife Carrie Symonds.
Why Melbourne couple were in Queensland
The married couple have both tested positive to COVID-19, triggering a surge for testing and vaccinations on the Sunshine Coast. At the time they travelled through NSW , stay-at-home orders were in place for travellers who'd been in Melbourne .
But Mr Cummings said in his blog post that the health secretary sought to blame NHS England chief executive Sir Simon Stevens, UK chancellor Rishi Sunak and the cabinet office for a "PPE disaster" in April past year.
"The lack of PPE killed NHS and care home staff in March-May", he writes.
Dominic Cummings, who has been engaged in a weeks-long feud with Downing Street, published a lengthy blogpost that he claimed showed details of how "Number10/Hancock have repeatedly lied about the failures last year".
"The PM is trying to influence officials/advisers to support the re-writing of history and is encouraging ministers to give false accounts to Parliament", he writes.
In his testimony, Cummings also lashed out at Johnson, saying he was "unfit for the job" of prime minister and that tens of thousands of people had died because of delays in ordering a succession of lockdowns.
Mr Cummmings also claimed Mr Johnson has a "clear plan" to leave Downing Street "at the latest a couple of years after the next election", which is scheduled for 2024.
But with the Conservatives riding high in the opinion polls, there is no incentive for the party's MPs to move against Mr Johnson, despite the "systemic incompetence" surrounding him, he adds.
"With this evidence that even the PM thinks Hancock is useless, why has he left him in charge?"
Johnson replied: "Totally f*****g hopeless".
Mr Cummings published the alleged leaks on Substack, an online platform that allows people to charge for newsletters.