The death toll among frontline health and care workers has surpassed 100, according to figures from the PA news agency.
Sir Keir Starmer said "too many" frontline workers have died during the fight against coronavirus "and we owe them a huge debt".
In a video message, the Labour leader said: "Nobody should put their lives at risk because they haven’t got the right protective equipment."
"We owe it to them to make sure that we’ve got the right equipment, in the right place at the right time, and we will continue to press on that," he added.
Intisar Chowdhury, the son of Dr Abdul Mabud Chowdhury, who died earlier this month after being diagnosed with Covid-19, challenged the Health Secretary during a live phone in on LBC.
Five days before he was admitted to hospital, Dr Chowdhury, 53, wrote a Facebook post asking Prime Minister Boris Johnson to urgently provide every NHS worker with PPE.
"We took very, very seriously what your father said and we’ve been working around the clock to ensure that there’s enough protective equipment and in the case of anybody who works in the NHS or in social care and has died from coronavirus we look into it in each case to find out the reasons where they might have caught it and what lessons we can learn," Mr Hancock said.
Mr Hancock said the Government has distributed more than a billion items of PPE.
He said: "People need to have the PPE according to the guidelines, the guidelines do say that in some cases PPE can safely be reused and that’s a good thing because PPE is in global short supply."
He was also quizzed about the findings of an unpublished 2016 pandemic drill, Exercise Cygnus.
He said: "It was before I was Health Secretary but I asked my officials to go back when this first came up in the press a few weeks ago and check that everything that was recommended was done and that’s the assurance I got, everything that was appropriate to do was done, that’s what I was told."
Mr Hancock added: "The preparations that we had in this country were amongst the most extensive in the world but of course you can’t prepare for a virus that is itself completely new by its nature."
Mr Hancock was asked by host Nick Ferrari whether he accepted that mistakes were made in the provision of PPE.
He replied: "Well, there are things that we’ve changed as we’ve gone through, both because we’ve learnt more things about the virus, also because things didn’t work out as we expected."
Coronavirus: Everything you need to know