The crisis that has engulfed the government over Dominic Cummings’ conduct continued to rage unabated after an unprecedented press conference in which the prime minister’s chief aide repeatedly refused to resign or apologise for breaking lockdown rules. After an outpouring of public anger rattled No 10, he attempted to explain why he drove 264 miles from London to his parents’ estate in Durham despite suspecting that both he and his wife had coronavirus. No 10 had hoped the move would draw a line under fury about Cummings’ behaviour after at least 20 Tory MPs called for him to quit and senior scientists accused him of undermining public health advice. But his appearance in the Rose Garden of No 10 raised yet more questions after Cummings admitted he had suspected both he and his wife had coronavirus when they made the decision to travel across the country with their son. During the lengthy press conference, Cummings claimed he drove to Durham because he needed possible back-up childcare from his teenage niece. He insisted he and his wife and child stayed in a separate building and communicated with his parents by shouting from a distance. He also admitted they made a separate trip after his family’s 14-day period of isolation to Barnard Castle, a beauty spot 30 miles away from Durham, which he claimed was necessary to check his eyesight was good enough for the longer drive back to London. He acknowledged they got out of the car and sat for 15 minutes by a river at a time when the lockdown rules banned nonessential trips, with exceptions for shopping, exercise and picking up medication. Cummings did, however, deny going back to Durham any time after his return to London on 14 April, saying witnesses who claimed they saw him in a bluebell wood on 19 April were mistaken. He said he had walked in woodland during his isolation period but only on his father’s property. Both the trip from London and to Barnard Castle would appear to be breaches of the government’s lockdown rules. Refusing to apologise and saying he had not offered to quit his post, Cummings said: “I don’t regret what I did … reasonable people might disagree.” Following the press conference, Johnson once again gave his full backing to his senior adviser, who was a key architect of Brexit and of Johnson’s election victory last year. The prime minister took a tone of contrition at some points, saying he did “of course regret the confusion the anger and the pain that people feel … as a country that has been going through tremendous difficulties and sufferings”. However, he also claimed that Cummings had fully explained himself and done nothing wrong, insisting: “I don’t think anyone in No 10 has done anything to undermine our messaging.” He insisted it was “very, very plausible” for Cummings to have gone to Barnard Castle because of problems with his eyesight. Johnson added: “I’m finding that I have to wear spectacles for the first time in years, because, I think, of the likely effects of this thing.” Numerous Tory MPs and cabinet ministers tweeted their backing for Cummings after the appearance but others remained unhappy with the explanation and feared that their constituents’ anger would not be assuaged. In other developments: The police and crime commissioner for Durham, Steve White, formally asked the chief constable of the force he oversees to launch an investigation into Cummings. The force said it was considering complaints, and it issued a statement clarifying that it had not given Cummings’ family advice about the lockdown, but had spoken to his father about security, contradicting an earlier statement. The retired chemistry teacher who first revealed Cummings’ trip to Barnard Castle said he should resign. After hearing the prime minister’s chief adviser’s defence of the journey, Robin Lees said: “I don’t think that was in the rules.” Some scientists warned that public loss of trust in the official advice could severely damage the test-and-trace strategy that is vital to managing the coronavirus epidemic and may lead to a new spike in infections. Robert West, a professor of health psychology at University College London’s Institute of Epidemiology and Health, said: “There are so many holes in his narrative that only the most desperate people who want to believe him could do so.” Johnson held a cabinet meeting about ending the lockdown, as the government tried to get its coronavirus strategy back on track in the wake of the Cummings allegations. At the government’s daily press conference, the prime minister said open-air markets and car showrooms would be able to open from 1 June and other nonessential retail outlets would be able to restart from 15 June. The chair of the Police Federation of England and Wales warned people not to get behind the wheel if they feel unwell or their eyesight is impaired. Cummings’ movements during the end of March and early April in breach of the lockdown were first disclosed in a joint investigation by the Guardian and Mirror. In his press conference, Cummings made clear that he had “not considered” resigning but acknowledged he could have acted differently by telling the prime minister he intended to travel to Durham before doing so. Insisting he had acted within the spirit and letter of the rules, he said: “The rules make clear that when dealing with small children, that can be exceptional circumstances and I think that was exceptional circumstances.” Asked why he was not resigning, he said: “There is understandable anger but a lot of that anger is based on reports in the media that have not been true. It’s extremely regrettable that the media were told some of these things were wrong and reported them anyway.” No 10 was repeatedly asked for a response hours in advance of the Guardian and Mirror’s original stories and declined to comment. Cummings revealed that his son needed to go to hospital while they were in Durham. The child was taken to hospital by ambulance along with his wife, who stayed overnight. Cummings left isolation the next day to pick up his wife and child from the hospital. His son later tested negative for coronavirus. Cummings said he had no physical contact with his parents but they had “shouted conversations at a distance”. He said the trip to Durham was not mentioned in articles for the Spectator written by him and his wife about their lockdown experience because he was worried about his security. In another remarkable admission, he said he had initially gone home from work fearing that his wife had contracted coronavirus, but then headed back into No 10 later that day to continue working, instead of isolating at home for 14 days. Later that evening he made the decision to travel to Durham to seek backup childcare, fearing that they would both become ill. In the end they did not have contact with other relatives, as his wife, Mary Wakefield, was well enough to care for their son. Cummings said he had not told the prime minister where he was going as Johnson had “a million things on his plate”, but he acknowledged this could have been a mistake. Earlier on Monday, more than 20 Conservative MPs called for Cummings to resign for breaking the lockdown, including the former ministers Tim Loughton and Steve Baker. But other Tory MPs appeared to be engaging in an effort by the party’s whips to dampen down public anger by issuing nearly identical cut-and-paste messages asking constituents to “rest assured” they were passing on concerns about Cummings to the “relevant colleagues”. Opposition leaders are to meet on Tuesday morning to discuss the next steps in holding Johnson and Cummings to account. Labour said the senior No 10 aide’s statement was not good enough. A party spokesman said: “The British people were looking for at least an apology from Dominic Cummings for breaking the lockdown. They got none. “The message from this government is clear: it’s one rule for Boris Johnson’s closest adviser, another for everybody else.” The Liberal Democrats and Scottish National party went further, calling for Cummings to be sacked. Ian Blackford, the SNP leader at Westminster, said: “What should have been a resignation statement turned out to be a botched PR exercise that changes nothing. It is now beyond doubt Dominic Cummings broke multiple lockdown rules.”