In testimony to lawmakers on Wednesday, Cummings accused Hancock of lying to the public and said he "should have been fired" for mistakes including testing failures that saw patients with the virus discharged from hospitals to nursing homes.
Pressed about the level of preparedness in government for the Covid-19 outbreak in February and March 2020, Mr Gove told MPs: "It was the nature of the coronavirus pandemic, a novel virus, that meant we had to adjust to the situation and while of course there are important lessons to be learned, and of course there were mistakes that we made, I would say two things".
Hancock said "the unsubstantiated allegations around honesty are not true".
"I'm just off to drive forward the vaccine programme, and then I'll be going to the House of Commons, and I'll answer questions there", he told reporters outside his home.
Mr Johnson denied Mr Cummings' central charge that British government failings had resulted in tens of thousands of unnecessary deaths.
He said Johnson was "unfit for the job" of prime minister.
"This has been an incredibly hard series of decisions, none of which we've taken lightly", he said.
But the prime minister did not answer when he was asked whether he has confidence in health secretary Matt Hancock who was subject to the worst of Mr Cummings' criticisms.
Asked if Mr Johnson regretted hiring Mr Cummings, the Prime Minister's official spokesman said: "It is a matter of fact that the Prime Minister took on Dominic Cummings into that role, into a senior role in No 10". "He told people he wanted to have open discussions where people could challenge policies, but when people did speak up he got rid of them".
"We put £1.4 billion extra into infection control within care homes, we established a care homes action plan, I remember very clearly, to ensure that we tried to stop infection between care homes", Mr Johnson said.
With nearly 128,000 deaths, the United Kingdom has the world's fifth highest official Covid-19 toll, far higher than the government's initial worst-case estimates of 20,000.
A mass vaccination campaign that started in December has brought confirmed cases and deaths down sharply, though Britain is now reckoning with a more transmissible new strain of the virus first identified in India.
One of the most damning allegations from Cummings was that Hancock's statement that the government had thrown a "protective ring around" care homes at the start of the pandemic was nonsense, and that instead people had been sent back from hospital who had contracted the coronavirus. It is spreading across the country and scientists expect it to become the dominant variant in Britain, but they say existing vaccines appear to work against it.
The government says it will begin an independent public inquiry into its handling of the pandemic within the next year.
Nearly three-quarters of British adults have had one dose of a coronavirus vaccine and 45 per cent have had both doses. Johnson said removing the remaining measures would depend on how much the new variant drove an increase in cases, and the speed of the vaccination campaign.
"We take decisions as fast as we can but we have to weigh the data", he said.
Asked whether the rise in the Indian variant Johnson said he did not "see anything now in the data" to divert from the June 21 target for the next stage of exiting lockdown but "we may need to wait" for more data.