Mr Cummings had previously accused the health secretary of telling Mr Johnson and others that people were being tested for Covid-19 before they were transferred into care homes when he knew that claim wasn’t true.
Appearing before a joint health and social care committee and science and technology committee hearing on Thursday morning, Mr Hancock also dismissed the suggestion that he was a liar who should have been sacked 15 to 20 times.
He said: “It is telling that no evidence has been provided yet. But there is a reason for that I think.
"Throughout this I have gotten out of bed in the morning with the view and the attitude that my job is to do everything I can to protect lives and get the country out of the pandemic.
“I have tried to do that with an approach of honesty and integrity, and, critically, answering questions in public and in private to the best of my ability.
"Sometimes you have to say you don't know because you are operating in a world where there are huge judgements being made with imperfect information, often at great pace."
The senior Tory told MPs he had “no idea” why Mr Cummings had a dispute with him, but had later become aware that he had wanted him fired.
He also struck back at Mr Cummings by claiming the UK Government was better without him.
Asked if he knew the former adviser wanted him to be fired, Mr Hancock added: “Yes, because he briefed the newspapers at the time. Or somebody briefed the newspapers. I now have a better idea who that was.
“I think the best thing to say about this, and this will be corroborated by lots of people in government, the best thing to say is that government has operated better in the past six months.”
Mr Hancock was also challenged about allegations that he used the “following the science” line as a way to blame scientists if things went wrong.
Refusing to deny it outright, he said: “My approach throughout has been that we are guided by the science. I try not to say that we follow the science.
“There are examples where ministers make decisions different to the scientific advice – one example is that when we brought back people from Wuhan in January, I was advised that they should be asked to go home and isolate and I said ‘no, they need to quarantine’.
“When it comes to the decisions around lockdown, we did accept and implement the scientific advice.
“I take full responsibility for the decisions that not only I take, but that are taken in my name as secretary of state and across the health family, and the NHS, Public Health England, in the department, and I know the Prime Minister feels very strongly the same.
“But of course you’re guided by the science.”
Mr Hancock also told MPs he was aware at the start of the pandemic that 820,000 people could die from Covid-19, but that ordering an earlier lockdown was against scientific advice.
He said: “The clear scientific advice at the time was that there was a need to have these tools like lockdown at your disposal, but also that the consequences and the costs of lockdown start immediately and, critically, the clear advice at the time was that there’s only a limited period that people would put up with it, would put up with lockdown.
“Now that proved actually to be wrong.”
Asked if Mr Johnson had had a conversation with him in April last year expressing surprise about untested people being discharged to care homes – as alleged by Mr Cummings – Mr Hancock said: “Not that I can remember.”
On discharging untested people, he explained the “challenge was not just that we didn’t have the testing capacity”, but there were concerns that people would be falsely told they did not have the disease.
Mr Hancock added: “At the same time, the clinicians were worried that, because it took four days to turn a test around, that if they leave somebody in hospital for those four days they might catch Covid and therefore go back to a care home with a negative result, but having caught it.”