The Health Secretary has urged the "small minority" of people who are breaking social distancing rules to change their behaviour, saying: "The more people follow the rules then the faster we will all be through this."
Matt Hancock, who said he understood how difficult a time the country was having because he has lost two people he was "fond of", added that the Government was "not planning any changes" imminently to the rules on exercising.
Earlier, Mr Hancock warned outdoor exercise could be banned if people continued to flout rules, as the coronavirus death toll rose sharply again.
It came ahead of a rare address from the Queen to the nation, in which she will urge the country to prove that this generation is "as strong as any".
In a televised message to be broadcast on Sunday evening, the head of state will recognise the pain felt by many families living through this "time of disruption".
She will personally thank frontline NHS staff, care workers and others carrying out essential roles for their efforts, in what is expected to be a deeply personal message reflecting her experience in other difficult times.
Speaking during the daily press conference on Sunday afternoon, as the country prepares to enter its third week in lockdown, Mr Hancock warned people not to flout rules around social distancing and said tougher restrictions could follow.
He added: "Following these rules is mission-critical if we're to protect the NHS, slow the spread and give the NHS time to expand capacity and so save lives.
"The more people follow the rules then the faster we will all be through this.
"So I say this to the small minority of people who are breaking the rules or pushing the boundaries: you're risking your own life and the lives of others and you're making it harder for us all."
Mr Hancock said the Government is not currently looking at tightening the rules on exercising outdoors, after images of crowded parks over the weekend sparked public outrage.
"What we are doing is being absolutely clear that the current rules must be followed," he said.
He said the rules are designed to include exercise to protect people's physical and mental wellbeing.
Earlier, the Met Office confirmed Sunday as the warmest day of the year so far, with both Heathrow and Kew Gardens recording temperatures of 19.8C at 1pm.
Speaking at the press conference, Dr Jenny Harries, the deputy chief medical officer for England, said: "It is not just what you are doing but how you are doing it.
"If you are sitting on a park bench, people tend to accumulate – it is very difficult to prevent that.
"Having rules where we are getting all of the benefits and minimising the risks and harms is an important approach to maintain.
"We have set those rules, we are enforcing against those rules and we will reiterate those rules, because that is the best way to be able to bend the curve down and stop the spread of the virus."
Mr Hancock, who returned from self-isolation on Friday after being struck ill with Covid-19, said he also offered his "profound sympathies" to the families and friends of those who have died.
He added: "I've lost two people that I was fond of so I understand what a difficult time this is for the country."
The Department of Health said the number of coronavirus-related hospital deaths stood at 4,934 patients as of 5pm on Saturday, up by 621 from 4,313 the day before.
In other developments:
– Police reprimanded Scotland's chief medical officer Dr Catherine Calderwood for visiting her family's second home, against her own advice that all Scots should stay home.
– Professor Stephen Powis, NHS England's national medical director, warned that while new cases of coronavirus appear to have stabilised, now is not the time to "take our foot off the pedal".
– In Wales, a further 12 people died after testing positive for Covid-19, bringing the total number of deaths to 166, Public Health Wales (PHW) said.
– Ireland's Health Minister Simon Harris said the Irish Government was aiming to almost double coronavirus testing to 4,500 a day.
– John Alagos, a 23-year-old NHS nurse, died after treating patients with coronavirus, his mother told the Mail on Sunday.
The eccentric Lord Bath of Longleat also died, aged 87, after testing positive for coronavirus.
Statement from Longleat and the Thynn family. pic.twitter.com/iJAszsmAD9
— Longleat (@Longleat) April 5, 2020
– Two staff members at north London's Pentonville Prison died after suffering Covid-19 symptoms, the Prison Officers' Association (POA) said, while the Ministry of Justice said hundreds of risk-assessed prisoners within two months of their release date would be temporarily sent home.
– Newly elected Labour leader Sir Keir Starmer accused the Government of making "serious mistakes" in its response to the coronavirus crisis, but pledged to engage "constructively" with ministers.
– And Carrie Symonds, Mr Johnson's pregnant fiancee, said she had spent the last week in bed suffering coronavirus symptoms, but was "on the mend".
The Queen will say in her address to the country and Commonwealth: "I hope in the years to come everyone will be able to take pride in how they responded to this challenge.
"And those who come after us will say that the Britons of this generation were as strong as any.
"That the attributes of self-discipline, of quiet, good-humoured resolve and of fellow feeling still characterise this country."
She will acknowledge the "grief" some have experienced, the "financial difficulties" many face, and the "enormous changes" the country is enduring, after almost two weeks of lockdown to tackle the spread of Covid-19.
With hundreds of thousands answering the call for NHS volunteers and others supporting vulnerable people in their communities, the monarch will say she hopes in the future everyone will be able to feel "pride" in how they rose to the challenge.
Commenting on the difficulties facing the nation, the Queen, 93, will say: "I am speaking to you at what I know is an increasingly challenging time.
"A time of disruption in the life of our country: a disruption that has brought grief to some, financial difficulties to many, and enormous changes to the daily lives of us all."