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Speaking to Sky News, Hancock said a full investigation would be carried out to see how working on the front line of the virus may have affected those health workers' safety.
While he stated he was not aware of any link between lack of personal protective equipment (PPE) and the deaths, Hancock said PPE was a 'precious resource'.
The Health Secretary said: "My heart goes out to their families... the fact that such a high proportion are from people that moved to this country to work for the NHS is really heart-rending... to see people who have literally given their lives for the NHS who were moved here to give that service and I think we should pay tribute to them."
Hancock said that although he would not criticise people for wearing more PPE than necessary, health workers should use it 'appropriately' - having advised during Friday's daily press briefing that staff only wear exactly the right amount of PPE, as distributing it has proven to be a 'herculean effort'.
At the briefing, Hancock also assured all social care and NHS staff that they will be able to get the testing they need for Covid-19, saying: "We have capacity for all key social care staff and NHS staff who need to get tested to get those tests."
With the Easter weekend now underway, he also urged people to resist the desire to go outside in groups, and to stay home.
Calling on the public to play their part in a 'national effort', he said: "However warm the weather, however tempting the beach or park, we need everyone to stay at home.
"NHS staff are battling day and night to keep desperately sick people breathing and they need you to stay at home."
He added: "Even if you are not directly involved... there is something that every one of us can do to play your part in the national effort.
"Stay at home, because spreading the virus today risks lives tomorrow and increases pressure of those working in the NHS.
"Do it for them. For it for the people you love."
It's okay to not panic. LADbible and UNILAD's aim with our Coronavirus campaign, Cutting Through, is to provide our community with facts and stories from the people who are either qualified to comment or have experienced first-hand the situation we're facing. For more information from the World Health Organisation on Coronavirus, click here.