The cabinet is to meet later to discuss the further easing of the coronavirus lockdown as Boris Johnson continues to be engulfed by claims his senior aide Dominic Cummings breached the rules.
The prime minister is this week expected to set out details of plans to lift restrictions.
But some Tory backbench MPs say the row has undermined the government’s message and want Cummings to resign.
Johnson has defended his aide, who had travelled from London to Durham.
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Speaking at Sunday’s Downing Street briefing, the prime minister said he believed Cummings had “no alternative” but to make the journey at the end of March for childcare “when both he and his wife were about to be incapacitated by coronavirus”.
At the press conference, Johnson also confirmed the phased reopening of England’s primary schools will begin on 1 June.
The plans to be outlined by the government will reportedly include information about the reopening of some non-essential shops in June.
However, BBC political correspondent Iain Watson said Johnson is finding it difficult to shift the political focus away from his key adviser.
Those criticising the prime minister include scientists on key government committees.
Stephen Reicher, a professor of social psychology who has advised the government on behavioural science during the pandemic, said the prime minister’s backing of Mr Cummings made him feel “dismay”.
He said trust was vital to maintaining public health measures. “You can’t have trust if people have a sense of them and us, that there’s one rule for them and another rule for us,” he told the BBC.
Two Church of England bishops have also criticised the prime minister.
The Bishop of Leeds, the Right Reverend Nick Baines, said Johnson was treating people “as mugs” and the Bishop of Bristol, the Right Reverend Vivienne Faull, accused the prime minister of having “no respect for people”.
Labour has called for an urgent inquiry into the allegations against Cummings.
Meanwhile, an investigation has been launched into a tweet posted on the official UK Civil Service Twitter account, which asked: “Can you imagine having to work with these truth twisters?”. The since-deleted message appeared shortly after Johnson’s daily news briefing.
The prime minister said he held “extensive” discussions on Sunday with Cummings, who he said “followed the instincts of every father and every parent – and I do not mark him down for that”.
The Observer and Sunday Mirror have reported two further allegations of lockdown breaches by his aide, although Johnson called “some” of the claims “palpably false”.
One report alleges that a witness saw Cummings in Barnard Castle, more than 25 miles from Durham, where he had been self-isolating, on 12 April.
Labour leader Sir Keir Starmer said he would have sacked Mr Cummings if he were prime minister, and he said Johnson’s failure to take action was “an insult to sacrifices made by the British people”.
Sir Keir said: “This was a huge test of the prime minister and he has just failed that test.
“Millions of people across the country have made the most agonising choices – not visiting relatives, not going to funerals – they deserve better answers than they got from the prime minister.”
Ministers to publicly support Cummings include the Health Secretary Matt Hancock and Cabinet Office Minister Michael Gove.
But Sir Roger Gale is among the Conservative backbench MPs to have publicly questioned Cummings’ position, telling BBC Radio 5 Live: “It’s up to the prime minister to exercise judgement about who he has around him. In this case, I do think that that judgement is flawed.”
Another Tory, former minister Paul Maynard, said of Cummings: “It is not as if he was unfamiliar with the guidance he himself helped draw up. It seems to me to be utterly indefensible and his position wholly untenable.”
Meanwhile, in a statement posted on Twitter, Amanda Hopgood, leader of the Liberal Democrats on Durham County Council, said: “a number of local residents have reported seeing Dominic Cummings on several occasions in April and May”.
She said that “given the clear public interest” she has referred the matter to Durham Police to see if there had been a breach of the coronavirus regulations.