Hancock denied rumours that the delays would mean no adults would get a first dose of the vaccine in April, but said it was important to make sure there was enough vaccine to give people a second dose within 12 weeks of their first.
But he acknowledged that in the short term, the country would receive fewer vaccines than planned a week ago, in part because of a shortfall from India's Serum Institute.
Vaccines have yet to be officially rolled out in England to the general population under 56, but a growing minority of people in this age group are likely to have been offered a dose due to their job or because they have a particular health condition.
The shortages come after Britain announced a new milestone Wednesday - more than 25 million people have now received a first shot of the vaccines developed by Pfizer Inc. and partner BioNTech SE or by Astra and the University of Oxford.
The Telegraph quoted a source stating that the second batch of five million doses that the SII is due to give the United Kingdom will only be delivered once it was green-lighted by the Indian government, which is deliberating on how to combat the renascence of the pandemic.
Reduced numbers of vaccines are due to problems with worldwide supplies, a government minister has suggested - but he insisted the issues would not slow down the easing of lockdown.
"It was disappointing news when we heard yesterday that the supplies weren't going to be available during April", he told the BBC.
Mr Jenrick said supply issues in the weeks ahead "isn't anything people should be anxious about".
Cabinet minister Robert Jenrick conceded that the rollout of vaccines would be slower than expected because of the shortage, while Professor Adam Finn, a member of the Joint Committee on Vaccination and Immunisation (JCVI), said vaccination of those aged under 50 "may kick off slightly later than we'd optimistically hoped". This type of clot, which can occur naturally, has been reported in fewer than 1 in every million people vaccinated and no causal link to the vaccine has been established, the MHRA said.
The Prime Minister's official spokesman said: "I would point back to what the Serum institute have said and the fact that they are one of the manufacturers of the Oxford/AstraZeneca vaccine".
India's foreign ministry says vaccine exports would be calibrated to domestic production, "as well as requirements of our national vaccine program".
Conservative former health secretary Jeremy Hunt questioned Mr Hancock over suggestions that vaccine exports could be blocked by the European Union, following comments by the head of the bloc's executive, Ursula von der Leyen.
Of this number, 21,886,125 were the first dose of a vaccine, a rise of 392,769 on the previous day, while 1,315,341 were a second dose, an increase of 90,906. It could have knock-on effects that last for months, including potential delays in lifting COVID-19 restrictions, he said.
Hancock said that European Commission head Ursula von der Leyen should respect contract law and that Britain expected to get the deliveries it had ordered.
Hancock told MP, "There will be no weeks in April with no first doses".
"There are very significant consequences to breaking contract law", Hancock said.