Boris Johnson has warned it may be "bumpy through to Christmas" and beyond as the UK deals with coronavirus.
Speaking to the BBC's Andrew Marr, the PM said there was "hope" in beating Covid, and 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.
It comes as a further 22,961 UK cases are reported, as previously unreported cases are added amid a technical issue.
Public Health England said an investigation into a technical glitch with the government's coronavirus dashboard identified 15,841 cases which were not included in the daily reports between 25 September and 2 October.
It said more than 75% of those cases - 11,968 - should have been reported between 30 September and 2 October.
"Every one of these cases received their Covid-19 test result as normal and all those who tested positive were advised to self-isolate," Public Health England's interim chief executive Michael Brodie said.
"NHS Test and Trace and PHE have worked to quickly resolve the issue and transferred all outstanding cases immediately into the NHS Test and Trace contact tracing system."
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."
Mr 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 prime minister 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'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."
Labour has been a long-standing critic of the performance of the Test and Trace system, with its leader, Sir Keir Starmer, accusing the government of having "lost control" of the virus.
Mr 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.
Is Boris Johnson optimistic or pessimistic about the fight against Covid? He seems to be a bit of both.
The prime minister's prediction for the next few months is gloomy - winter will be "bumpy".
Restrictions of varying degrees could be with us for months.
But at the same time, he is urging people to be fearless if they use common sense - and believes a "radically different" spring is around the corner.
The short-term forecast is downbeat. But the PM wants to retain hope that things could improve again in the not so distant future.
Mr 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".
Mr 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".
Mr 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 told Andrew Marr.
"I hope there isn't. But until the government publish clear guidelines, that suspicion will always linger."
Mr 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 prime minister 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.
- DO YOU HAVE 'LAPTOP SYNDROME'?: Tips to avoid this condition if you're working from home
- LOVE LIFE: Fancy an escape? Watch the addictive new series starring Anna Kendrick