Mrs May failed to answer four direct questions on her knowledge of the test, which reportedly took place weeks before MPs approved the £40 billion Trident renewal programme in July.
Reports in The Sunday Times claim that the launch of an unarmed Trident II D5 missile from a British submarine off the coast of Florida in June malfunctioned.
Mrs May did not mention this test in a speech to MPs before the Commons vote, in which she urged them to back renewal, leading to allegations the malfunction has been covered up.
Labour leader Jeremy Corbyn called the failed test “a pretty catastrophic error”, while the SNP’s Nicola Sturgeon has called for “full disclosure” about who knew what when.
Labour peer and former senior Royal Navy officer Admiral Lord West added it was “bizarre and stupid” to not tell anyone about the test.
Appearing on BBC One’s Andrew Marr Show, Mrs May said she had “absolute faith” in the Trident missiles.
However, she repeatedly failed to address whether she knew about the failed test before her speech to MPs.
She said: “When I made that speech in the House of Commons, what we were talking about was whether or not we should renew our Trident, whether or not we should have Trident missiles, an independent nuclear deterrent in the future.
“I think we should defend our country, I think we should play our role in Nato with an independent nuclear deterrent.
“Jeremy Corbyn thinks differently, Jeremy Corbyn thinks we shouldn’t defend our country.”
She gave similar answers to further questions on the subject.
Mr Corbyn seized on suggestions the missile veered away from its intended target near Africa.
The Labour leader told Sky News’ Sophy Ridge on Sunday: “I think this failure is something that ought to pause everyone for a moment and just think what happened.
“We understand the Prime Minister chose not to inform Parliament about this and it’s come out through the media some months later.
“It’s a pretty catastrophic error when a missile goes in the wrong direction and whilst it wasn’t armed, goodness knows what the consequence of that could have been, I think we need a serious discussion about that.”
SNP leader Ms Sturgeon, meanwhile, tweeted: “This is a hugely serious issue.
“There should be full disclosure of what happened, who knew what/when, and why the House of Commons wasn’t told.”
Admiral Lord West told BBC Radio 4’s The World this Weekend that the Government had “made a bit of a pageant” of previous missile tests.
He said: “From what the Government says there was a minor glitch with the missile and they’re quite happy with the missile.
“In which case go ahead and let people know, otherwise we’re a bit rather like the Soviet Union used to be, or like North Korea or China, where they won’t admit to things going wrong when you’re actually testing them to see if they do or don’t go wrong.”
He added: “I think it is bizarre and stupid that they didn’t say that there’d been a firing and that there had been a missile malfunction and that it was a minor fault.”
The Sunday Times said the cause of the failure remained top secret but questions could be asked over the Government’s failure to publicise the failed test
Labour MP and former defence minister Kevan Jones told the newspaper: “The UK’s independent nuclear deterrent is a vital cornerstone for the nation’s defence.
“If there are problems, they should not have been covered up in this ham-fisted way.
“Ministers should come clean if there are problems and there should be an urgent inquiry into what happened.”
A Government spokesman said: “The capability and effectiveness of the Trident missile, should we ever need to employ it, is unquestionable.
“In June the Royal Navy conducted a routine unarmed Trident missile test launch from HMS Vengeance, as part of an operation which is designed to certify the submarine and its crew.
“Vengeance and her crew were successfully tested and certified, allowing Vengeance to return into service. We have absolute confidence in our independent nuclear deterrent.
“We do not provide further details on submarine operations for obvious national security reasons.”
Nia Griffith, Labour’s Shadow defence secretary, said: “This report of a Trident missile veering off course during a test is clearly a very serious matter indeed, and we need to know exactly what happened.
“Furthermore, it is completely unacceptable that today the Prime Minister chose to side-step questions on the test, and would not even tell us when she knew about the incident.
“I am demanding the Prime Minister come to Parliament tomorrow to give a full explanation to MPs.”