The Boris Johnson said some of the “commentary” didn’t “bear any relation to reality”.
And Health Secretary Matt Hancock – who was accused by Mr Cummings of lying about his handling of the pandemic – said the allegations were “not true”.
But Labour accused him of failing to answer “pressing questions” on testing, PPE and care home mistakes.
In a letter to Mr Hancock, the party’s shadow health secretary Jonathan Ashworth said “it is extremely disrespectful to those who have died and their loved ones” not to address the claims.
Labour has urged the government to bring forward the public inquiry into the Covid pandemic – due to start next year – but this idea was rejected.
In total, 127,748 people have died in the UK within 28 days of a positive test for the disease, according to official figures.
During a scathing attack on Mr Johnson and Mr Hancock on Wednesday, Mr Cummings told a committee of MPs “tens of thousands of people died, who didn’t need to die”, as a result of government mistakes.
More than 42,000 of total UK deaths involving Covid have happened in care homes in England and Wales, according to the Office for National Statistics.
Mr Cummings claimed Mr Hancock had assured him and the prime minister early in the pandemic that people “were going to be tested before they went back to care homes [from hospitals]”.
“We only subsequently found out that that hadn’t happened,” said Mr Cummings, adding that subsequent government statements about putting a shield around care homes were “complete nonsense”.
At the Downing Street press conference on Thursday afternoon, Mr Hancock said “of course” he had committed to testing everyone returning to a care home from hospitals, but that it “took time” to build the testing capacity.
He said without the capacity, “it wasn’t possible” to test everyone, but that was why he set his target of 100,000 tests a day – something Mr Cummings had described as “stupid” and counter-productive.
Mr Cummings also attacked the health secretary’s integrity, saying he “should have been fired for at least 15 to 20 things, including lying to everybody on multiple occasions in meeting after meeting in the Cabinet Room and publicly”.
And he told the MPs Mr Johnson – for whom he worked until he was ousted in a Downing Street power struggle last autumn – was “unfit for the job” of leading the country.
He added that he had repeatedly urged Mr Johnson to sack Mr Hancock, calling the health secretary “completely incapable of doing the job”.
Mr Hancock hit back as he answered an urgent Labour question on Covid in the Commons, saying he wanted to “put formally on the record that these unsubstantiated allegations around honesty are not true,”
He added: “I’ve been straight with people in public and in private throughout.”
In an apparent barb aimed at Mr Cummings, Mr Hancock said the “operation and function” of government had become “easier” over the past six months.
On a visit to a hospital in Colchester, Essex, the prime minister, when asked if tens of thousands of people had died needlessly, replied: “No, no I don’t think so.
“Of course, this has been an incredibly difficult series of decisions, none of which we’ve taken lightly.”
He added that “at every stage we’ve been governed by a determination to protect life, to save life, to ensure NHS is not overwhelmed”.
But, on a visit to Avonmouth, Labour leader Sir Keir Starmer said that the “buck stops” with the prime minister over the government’s performance on Covid.
He added: “Dominic Cummings can’t be the last word on this. We need to get all the available evidence….
“I don’t think the prime minister has made good decisions in this pandemic.”