"Our UK domestic supply chain is not experiencing any disruption and there is no impact on our delivery schedule", said a spokesman for the firm, which makes the vaccine developed by the University of Oxford.
The number of vaccine supplies distributed to Wales is decided by the UK Government and has been based on population numbers.
Hancock told MPs in the Commons, "In April, supply is tighter than this month and we have a huge number of second doses to deliver".
"The impact of this shortage of supplies will happen on the group that we were hoping to start on in April, which is the people under the age of 50 without any pre-existing conditions, who are now going to have to wait until May", he told BBC Radio 4's Today programme.
Britain's medicines regulator said there had been five cases of a rare type of blood clot in the brain among 11 million people given AstraZeneca's vaccine but said that it found the benefits of the shot far outweighed any possible risks. Jab sites in England have now been ordered to stop vaccinating people under 50 during April.
"We will do all we can, and do everything necessary, to secure the supplies that are contractually committed to protecting people in this country", UK Health Secretary Matt Hancock said at a Downing Street briefing on Wednesday evening.
A delivery had been expected from the Serum Institute of India but has been held up by four weeks.
SII chief executive Adar Poonawalla told The Telegraph: "It is exclusively dependent on India and it has nothing to do with the SII".
Housing Secretary Mr Jenrick told BBC Breakfast the final goal of vaccinating all adults with one dose by the end of July was still on track.
"The vaccine roll-out will be slightly slower than we might have hoped but not slower than the target", he said.
"We're going to move forward as quickly as we possibly can but it won't be as fast as we might have hoped for a few weeks, but then we have every reason to believe that supply will increase in the months of May, June and July". "There will be no cancelled appointments as a result of supply issues, second doses will go ahead as planned".
Hancock said Britain expected doses of Moderna's vaccine to arrive "in the coming weeks".
"I've called time and time again for the publication of data on how many of each type of vaccine has been, and is being distributed to each United Kingdom nation".
"As a result, we will receive slightly fewer vaccines in April than in March, but that is still more than we received in February, and the supply we do have will still enable us to hit the targets we have set", he said.
Supply constraints are the biggest threat to Britain's vaccine rollout - now the swiftest among the world's major economies - and health officials warned that the programme would face a significant reduction in supplies from March 29.
England's Chief Medical Officer Chris Whitty said anecdotal reports suggested that some people had not turned up to vaccination appointments after the shot was suspended in some European countries, but record numbers were still being vaccinated.
NHS leaders said there will be a "significant reduction" in the weekly vaccine supply available from the week beginning March 29.
It has been confirmed that there has been delays of 5m doses of the Oxford/AstraZeneca vaccine from India which is in part why there will be fewer doses over the coming weeks.
AstraZeneca has partnered with the institute, which is the world's largest vaccine manufacturer, for supplies to the Indian government but also to other countries, including low and middle-income ones.