Boris Johnson promises to “listen and learn” but ex-leader Michael Howard calls for him to quit.
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Boris Johnson has admitted results from two by-elections were not “brilliant” but has vowed to keep going.
His party lost the Devon seat of Tiverton and Honiton to the Lib Dems, and Wakefield to Labour.
Mr Johnson said he would “listen and learn” and focus on “delivering for the people of this country”.
The two defeats prompted Conservative party chair Oliver Dowden to quit saying: “We cannot carry on with business as usual.”
And ex-Conservative leader Michael Howard has said the party and the country would be “better off” under new leadership.
Speaking to the BBC’s World at One, Lord Howard said the members of the cabinet should consider resigning and that party rules should be changed to allow for a new confidence vote in the prime minister.
The prime minister survived a vote on his leadership earlier this month, but 148 of his MPs voted to oust him, leaving him weakened. Under Tory party rules, it means that a further vote of confidence cannot be held for another year.
The by-elections took place as Mr Johnson faced questions over his leadership since stories emerged of Covid rule-breaking in Downing Street during the pandemic, along with concerns over rising costs and a squeeze on wages.
Mr Dowden announced his resignation in the early hours of Friday morning following the defeats, saying in a letter to the prime minister that supporters were “distressed and disappointed by recent events and I share their feelings”.
“Somebody must take responsibility and I have concluded that, in these circumstances, it would not be right for me to remain in office.”
Other cabinet ministers have rallied round the PM, who is currently in Rwanda attending a meeting of Commonwealth government leaders.
Taking questions at a press conference in Kigali, Mr Johnson said Mr Dowden did a “lot of good work” but added: “I genuinely, genuinely don’t think the way forward is to focus on issues of personality whether they are mine or others.
“I’m not going to pretend these are brilliant results,” he said adding: “We’ve got to listen, we’ve got to learn.
“There will still be some tough times ahead, no doubt people will continue to beat me up and say this or that to attack me.”
He said that was “fine” but that he would be focusing on “delivering for the people of this country”.
As you might expect the by-election defeats came up repeatedly during the prime minister’s press conference, but Boris Johnson would not accept any personal responsibility – or acknowledge that his leadership had any part to play.
He said he didn’t think British politics should be about personality – but for some of his own MPs his personality is exactly the problem.
Some talk about the need for a change of direction or as one put it to me earlier a “reset” moment, but there was little sign of that from the prime minister.
He put much of the blame on the cost of living, saying when people find it tough – they blame governments.
But while that’s undoubtedly a huge issue facing the country, for some Tories Boris Johnson’s style of government is an issue in itself.
The prime minister was defiant – talking about his plan and seeming keen to press on, a mood echoed by his team here in Kigali.
He suggested governments shouldn’t be defined by by-elections – pointing to administrations that have survived defeats in the past. But the way he responds to these by-election losses could well prove defining.
Cabinet ministers including Culture Secretary Nadine Dorries and Deputy Prime Minister Dominic Raab have given Mr Johnson their backing.
Asked if he had full confidence in the prime minister’s ability to win the next election, Health Secretary Sajid Javid said: “Boris Johnson is our leader, is the PM and he will be taking us into the next election.”
And Ben Houchen – Conservative mayor of Tees Valley – warned his colleagues against a “knee jerk reaction” and said that holding a leadership contest during a cost of living crisis would make the party look “ridiculous”.
In a statement, Trade Minister Penny Mordaunt said she was “disappointed” for the party but did not specifically mention support for the prime minister.
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In Tiverton and Honiton – where former MP Neil Parish quit after he was found watching pornography in Parliament – the Lib Dems won with a 30% swing.
The party quashed a Conservative majority of 24,239 – the largest ever to be overturned at a by-election.
Speaking from the Devon constituency, Liberal Democrat leader Sir Ed Davey said the Conservatives were “chaotic” and had “not got the policies for our country”.
The Conservatives also suffered a loss in West Yorkshire where Labour secured a 12.7% swing to regain the Wakefield constituency by 4,925 votes.
Labour leader Sir Keir Starmer said the election result was “a clear judgment on a Conservative Party that has run out of energy and ideas”.
Labour “is back on the side of working people, winning seats where we lost before, and ready for government,” he added.
