Speaking at a BBC program, the PM said there was "hope" in beating Covid, but called on the public to "act fearlessly but with common sense".

He said the government was taking a "balanced" approach between saving lives and protecting the economy.

His comments come as the UK announced more than 10,000 new cases for the first time since mass testing began.

There were 12,872 new cases announced on Saturday. However, the government said a technical issue meant some cases earlier in the week were not recorded at the time so had been included in Saturday's data.

Government sources said that because of an IT error there had been a delay entering the details of positive cases into the NHS Test and Trace system in England, which slowed the process of finding their contacts.

The problem was a "serious issue" but it has been "contained", sources said.

The Prime Minister said: "The best thing we can do now for all those who have suffered in the course of this pandemic is bring it to an end in the speediest possible way."

Labour's shadow health minister, Alex Norris, criticised the interview as a "wasted opportunity" to set out a "serious strategy to improve public confidence in the government's handling of this crisis".

He said: "Instead [the PM] waffled and ducked every question. His serial incompetence is holding Britain back."

Johnson said he believed over the "next few weeks and months" the "scientific equation will change whether that is vaccines or testing" and there will be "progress" in beating the virus.

As a result, he said there was "hope" and "things can be significantly different by Christmas", as well as being "radically different" by spring.

But, the PM warned there could be "a very tough winter for all of us", adding: "I tell you in all candour, it will continue to be bumpy through to Christmas and may even be bumpy beyond."

Labour has been a long-standing critic of the performance of the Test and Trace system, with its leader, Keir Starmer, accusing the government of having "lost control" of the virus.

Johnson said the system was "not perfect" and that he was "frustrated with it".

But he defended its "massive increase in capacity", saying it had "made a huge difference" in tackling Covid-19.

Johnson also stood by the Eat Out to Help Out restaurant discount introduced in August, which some critics have said added to the rise in coronavirus cases in September.

"In so far as that scheme may have helped to spread the virus then obviously we need to counteract that and we need to counteract that with the discipline and the measures that we're proposing," he said.

But he insisted it was "right to reopen the economy" as the government tries to "strike the right balance".

Johnson said he took "full responsibility for everything that has happened since the pandemic began".

Asked about how effective the latest local lockdowns were in tackling the growing number of cases, the prime minister said it was "too early to say".

Johnson said he understood the "frustrations" of people living in the affected areas - as well as a number of his own backbench MPs - but defended the action, saying: "I'm a freedom-loving Tory. I don't want to have to impose measures like this, are you crazy?

"This is the last thing we want to do. But I also have to save life. And that's our priority."

"And I also think, by the way, that's the priority of the British people and I think they will want to see their government continue to work, continuing to fight the virus and that's what we're doing."

Labour's shadow health secretary Jon Ashworth earlier called for the government to publish guidelines on what criteria they used to impose local lockdowns, and to involve local council leaders and health officials.

He said there were questions as to why current areas have extra restrictions, while the constituencies of cabinet ministers with higher case numbers remained unchanged.

"Because there are no clear guidelines as to why an area goes into restrictions, and how an area comes out of restrictions, then there is a suspicion that there is political interference," he said.

"I hope there isn't. But until the government publish clear guidelines, that suspicion will always linger."

Johnson was also asked about his health following his own experience of fighting coronavirus in March and April - and in light of US President Donald Trump testing positive.

The PM said when he had the virus he was "too fat" and it was a "teachable moment for our great country" to get on top of the issue of obesity.

But he claimed it was "balderdash" that he was still suffering from the effects of coronavirus, known as "long Covid".

He said the claim was "drivel", adding: "It is balderdash and nonsense. I can tell you I'm fitter than several butchers' dogs."

The Conservatives are currently holding their first virtual party conference due to coronavirus restrictions on mass gatherings.