The Health Secretary told a Downing Street news conference on Monday that the coronavirus outbreak is being brought “under control”.
Mr Hancock said: “Don’t blow it now. We are on the route out. We are protecting the most vulnerable. We are getting the virus under control,” he said.
“Together, I know that we can do it and we have got to stick at it.”
Asked whether families could feel confident in booking a break for the summer holidays as all adults may be vaccinated by the summer, Mr Hancock said: “On the summer holidays, I’m going to Cornwall and I’ve said before I think we’re going to have a great British summer, but we’ve got a lot of work to do between now and then.
“What we can do is see the line of sight to vaccinating everybody by September and anything before that would be a bonus.
“As the Foreign Secretary said yesterday, we’re driving this as fast as we possibly can.”
Government data for January 17 shows of the 4,514,602 jabs given in the UK so far, 4,062,501 were first doses – a rise of 225,407 on the previous day’s figures.
Some 452,301 were second doses, an increase of 2,565 on figures released the previous day.
In Northern Ireland, a total of 125,717 people have received a first dose so far, a daily rise of 5,330.
As the vaccination programme continues with the rollout of jabs for over-70s and the clinically vulnerable, Professor Stephen Powis warned it would be “some time” before the programme starts to ease pressures on hospitals.
Professor Powis said around 15,000 people had been admitted to hospitals with coronavirus since Christmas Eve.
NHS England’s national medical director said: “It is absolutely critical that we continue to stick to those social distancing rules that are in place. That we don’t rely yet on vaccines coming to our rescue.
“It will be some time before the effects of the vaccination programme are seen through into reducing pressure on hospitals. We all have a role to play in reducing the risk of transmission.”
It comes as the latest NHS England data revealed a record 34,336 Covid-19 patients were in hospital in England as of 8am on Monday.
The figure is up 7 per cent on a week ago, and up 94 per cent since Christmas Day.
In London, the number stands at a record 7,917 patients, up 1.5 per cent in a week, while south east England has 5,493 patients, up 0.5 per cent; Eastern England has 4,244, up 0.6 per cent.
Nearly 90,000 people have died of the virus in the UK since the start of the pandemic, after a further 599 coronavirus deaths were recorded on Monday.
Public Health England said there had been a further 37,535 lab-confirmed cases of coronavirus in the UK on Monday. It brings the total number of cases in the UK to 3,433,494.