The departure of the prime minister’s chief adviser Dominic Cummings is a chance to “reset government”, a senior Tory MP said.
Mr Cummings left Downing Street for the last time on Friday following internal battles about his role.
Former Brexit Secretary David Davis said Boris Johnson had taken “decisive action” in removing his aide.
The prime minister’s spokesman said Mr Johnson was not distracted by the row was “focused” on tackling coronavirus.
Mr Cummings and director of communications Lee Cain, who resigned on Thursday, will work out their notices at home following tensions within No 10.
Mr Davis said Mr Cummings had a “very confrontational-style” which had eventually turned people in Downing Street against him.
He said: “Lots of my colleagues are hoping for a new relationship – more openness and interaction with Parliament – and I am told the cabinet is hoping to get more say as it were in events.”
The departure of Mr Cummings and Mr Cain comes as the government grapples with the coronavirus pandemic, and as trade talks between the UK and the EU on their future relationship reach a “make or break” point.
Labour said the PM could “rearrange the deckchairs all he wants… but the responsibility for this government’s incompetence still lies firmly at Boris Johnson’s door”.
“The fact there is no plan and no focus in the government’s response to Covid is entirely down to him,” a party source said.
Mr Cummings had a notoriously difficult relationship with Conservative MPs, some of whom have said it is time for things to be done differently in Downing Street.
Former Tory leader Sir Iain Duncan Smith wrote in the Daily Telegraph that Mr Cummings’ influence had led to “a ramshackle operation in the hands of one man”.
Another MP said was it was time to “rejoice” at the departure of the two aides, the BBC’s deputy political editor Vicki Young reported.
Lord Gavin Barwell, who was former Prime Minister Theresa May’s chief of staff, told BBC Radio 4’s Today programme there was an opportunity for Mr Johnson to rebuild relations with Conservative MPs and “set a less confrontational and more unifying tone that is maybe more in tune with his natural instincts”.
And former Northern Ireland secretary, Theresa Villiers, said: “Both Dominic Cummings and Lee Cain were pretty dismissive of backbenchers and sometimes ministers and secretaries of state, and I don’t think that was helpful.
“I do think it’s important that whoever takes over has a different approach.”
Both Mr Cummings, 48, and Mr Cain, 39, are veterans of the Vote Leave campaign and worked closely with Mr Johnson to deliver the Brexit vote during the 2016 EU referendum.
Lord Edward Lister, who was Mr Johnson’s chief of staff when he was Mayor of London, will become interim chief of staff pending a wide-ranging shake-up of the prime minister’s team.
Mr Cain will be replaced by James Slack, who is currently the prime minister’s official spokesman.
By Nicholas Watt, Newsnight Political Editor
There are conflicting accounts of the final meeting involving Boris Johnson and his departing aides, Dominic Cummings and Lee Cain.
Newsnight has been told that relations between the trio “went off the cliff” in the early afternoon on Friday.
The prime minister’s team reportedly learned at around 14:00 GMT that Mr Cummings’ team had described him as indecisive.
They also heard of alleged briefing against the prime minister’s partner, Carrie Symonds, who had apparently been uneasy about a plan to promote Mr Cain to chief of staff.
Newsnight was told that the prime minister expressed his displeasure during a meeting with the two men. He reportedly told them he knew what they were up to and they would have to leave.
This account is strongly disputed by Mr Cain and Mr Cummings’ side.
A source said that Mr Cummings and Mr Cain held a very friendly and warm 45-minute meeting with the prime minister. Mr Johnson reportedly told them: “I want to get the band back before the next election.”
The prime minister then agreed to a request from Mr Cain to sign a pair of boxing gloves, used during the general election, emblazoned with the words: “Get Brexit Done.”
Mr Cummings left the building. But Mr Cain, the outgoing director of communications, remained for a farewell reception in the press office. That was addressed by the prime minister before Mr Cain was “banged out” in the style of a traditional newspaper farewell.
The prime minister reportedly said that relations remain good with the two aides but that relationships can break up, at which point you no longer live together. He was keen to emphasise there was no ill will, according to this account.
Mr Cummings prompted controversy this summer after it emerged he made a 260-mile trip from London to County Durham with his family at the height of the UK’s first coronavirus lockdown.
The adviser later said the journey was intended to secure childcare, but he was mocked for claiming a subsequent outing to the picturesque town of Barnard Castle was to help test his eyesight.
Jill Rutter, senior research fellow at the UK in a Changing Europe think tank, said Mr Cummings had been “disproportionately influential” but the year he had been in Downing Street would not have been long enough to carry out long-term reform.
She said: “Ultimately it is up to the prime minister who he appoints as his adviser, and advisers only get any authority from the prime minister, so you have to assume that what Dominic Cummings was doing was what the prime minister wanted him to do, up until he didn’t which is what appears to have happened yesterday.”
Mr Cain is said to have left Downing Street through a discreet exit on Friday evening.
But Mr Cummings walked through the black front door of No 10 with a cardboard box and was later seen arriving home with a bottle of champagne.
Mr Davis said his exit from Downing Street holding a cardboard box was “entirely deliberate” as he wanted to leave an “image”.