As well as the revelations about the PM's opinion of his health secretary, former chief adviser Cummings also published messages - dated 27 April 2020 - in which the PM appears to call the situation around personal protective equipment (PPE) "a disaster".
At his appearance last week before the Health and Social Care and Science Committees, Hancock said he had seen no evidence to suggest any medical staff had died because of a lack of PPE.
In his blog, Cummings accuses the health secretary of trying to blame NHS England chief executive Sir Simon Stevens, Chancellor Rishi Sunak and the Cabinet Office for last April's "PPE disaster".
"The lack of PPE killed NHS and care home staff in March-May," he writes.
As we reported earlier, Dominic Cummings - the PM's former chief adviser - has published expletive-laden messages apparently from Boris Johnson, in which Johnson brands the health secretary "hopeless".
In Guatemala, where Daniela is from and her family live, the daily figures for new Covid infections are currently at an all-time high.
"Here we have the NHS and so much access to get care if you get Covid," Daniela says.
He says after the health and economic consequences of coronavirus have been dealth with the government will have set a disastrous president in terms of the future of liberty.
Mark Harper, chair of the Covid Recovery Group, says his worry is that there will be a continual extending of restrictions.
Health Secretary Matt Hancock denies this is likely, saying eradicating the virus is not possible - and the government is aiming "to live with Covid, like we do with flu".
The four-week delay is intended to get a second jab into "a majority" of over-40s by 19 July and the pause is estimated "to save thousands of lives", says Hancock - but he adds that a further pause would not save many more lives.
Jonathan Ashworth says lifting restrictions now would be "akin to throwing petrol on a fire", confirming that as the shadow health secretary, he will be backing the delay to restrictions for a further four weeks.
He says the health secretary will be forever branded "Hopeless Hancock" - following Dominic Cummings' revelations of text messages from the PM earlier - a view which, Mr Ashworth adds, might now be shared by members of the public.
Ashworth says the government "failed to protect our borders" and he accuses them of giving the Delta variant "the red carpet" - reiterating Labour leader's Sir Keir Starmer's attack at PMQs earlier today.
Matt Hancock confirms - as reported earlier by the BBC - that vaccinations will be mandatory for older people's care home workers in England and there will be consultation over whether to do the same for NHS workers and domiciliary carers.
He says: "The vast majority of staff in care homes are already vaccinated but not all and we know that the vaccine not only protects you but it protects those around you.
"And therefore we will be taking forward the measures to ensure the mandation for as a condition of deployment for staff in care homes and we will consult on the same approach in the NHS in order to save lives and protect patients from the disease," he says.
He says there were "no proposals to go any wider" than those in domicillary care and the NHS.
Health Secretary Matt Hancock has opened the debate on whether to delay the easing of coronavirus restrictions.
Boris Johnson says delaying the planned lifting of restrictions until 19 July will give the NHS "a few more crucial weeks" to get people vaccinated.
Treasury Secretary Stephen Barclay says the delay in easing lockdown restrictions, announced on Monday, "presents additional challenges" to business.
BBC Radio 5 Live is hearing what listeners think about plans to make all care home staff have a Covid vaccine.
Tina Bentley, who runs a care home in Essex, says she is absolutely behind vaccinating all care home staff?
But another caller, Molly, says she thinks introducing compulsory vaccines in the care sector is a “slippery slope”
No 10 says Boris Johnson has full confidence in the health secretary after Mr Cummings' latest attack
The health secretary says it is a "sensible and reasonable step” and could be extended to the NHS
The government has said some restrictions will stay in place in England after 21 June