After more than a year as prime minister, Rishi Sunak, a Conservative, has failed to close a yawning gap in the polls. On Monday he did something new.Prime Minister Rishi Sunak of Britain fired one of his most senior and divisive ministers on Monday, in a reshuffle of his top team that unexpectedly brought a centrist predecessor, David Cameron, back into government.The departure of Suella Braverman as home secretary and the surprise return of Mr. Cameron as foreign secretary were the latest in a series of convulsions that have rocked the governing Conservative Party since the fateful Brexit referendum that Mr. Cameron called in 2016, and signaled the peril facing Mr. Sunak as he nears a general election expected next year.After 13 years in Downing Street, the Conservatives’ grip on power appears to be slipping, with the party trailing Labour by around 20 points in the polls against a challenging economic backdrop, with sluggish growth and inflation eroding living standards, and a public sector under acute strain after years of Conservative-led austerity.Mr. Sunak has tried various gambits to address his party’s unpopularity with voters, weakening environmental targets, pledging to defend motorists and promising tougher sentencing for serious criminals. None seem to have worked.At the same time, Ms. Braverman, who is seen as a rival within the party, had become increasingly emboldened as home secretary, raising her profile and appearing to prepare the ground for a leadership bid if the Conservatives lose the election as many expect.Last week she wrote an extraordinary opinion article in The Times of London, which was not authorized by Downing Street, in which she criticized the police for not seeking to ban a pro-Palestinian protest march in the capital, and described the demonstrators as “hate marchers” and “Islamists.”Protesters in solidarity with Palestinians in Gaza, on Vauxhall Bridge in London, on Saturday.Hollie Adams/ReutersAfter counterprotesters clashed with the police on Saturday, critics accused Ms. Braverman of inflaming tensions and encouraging far-right demonstrators onto the streets, and her position was judged untenable by Downing Street.Mr. Sunak and Ms. Braverman spoke by phone on Monday, and in the shuffle of jobs that followed her departure, she was replaced by the more emollient former foreign secretary, James Cleverly, freeing up his position for Mr. Cameron.Both men are regarded as moderates and the changes appeared to signal a shift away from the divisive politics that were championed by Ms. Braverman, whose focus on cultural issues had become a feature of Mr. Sunak’s government in recent months.Neither of the two appointments was good news for the right-wing faction of the Conservative Party where Ms. Braverman had a small but vocal group of supporters.Nor was Mr. Sunak’s decision to keep Jeremy Hunt as chancellor of the Exchequer. Mr. Hunt’s resistance to offering tax cuts has antagonized a wider group of Conservative lawmakers. He, like Mr. Cameron, campaigned against Brexit in 2016, but Mr. Hunt has made controlling inflation his priority and says that reducing taxes will have to wait.The return to the cabinet of Mr. Cameron may remind some voters of the political chaos that he triggered in 2016 when Britons ignored his recommendation and narrowly voted to leave the European Union. Mr. Sunak is the fourth Conservative leader to have become prime minister since Mr. Cameron stood aside after the referendum result, which sent shock waves around Europe.David Cameron, Britain’s new foreign secretary, departing 10 Downing Street on Monday.Andy Rain/EPA, via ShutterstockMr. Sunak restored some stability when he succeeded Liz Truss as prime minister last year, but his latest reshuffle risks reopening ideological divisions that have dogged the party in recent years. Though the salience of Brexit has faded in British politics, Mr. Cameron — who led the campaign against it — will now be partly responsible for promoting the policy around the globe.Yet, while bringing back Mr. Cameron is a political gamble, Mr. Sunak may have judged the risk worthwhile. He has limited time to win back voters, or possibly even to limit the scale of a defeat in the looming election.Ms. Braverman had lost her job as home secretary once before, under the short-lived government of Ms. Truss, but she was given it back by Mr. Sunak when he entered Downing Street. She used her position in cabinet to push hard-right policies and embraced polarizing rhetoric, describing migration as a “hurricane,” the arrival of asylum seekers on the British coast as an “invasion” and homelessness as a “lifestyle choice.”While Mr. Sunak’s language was more measured, he supported most of her ideas — in particular, her pursuit of a policy to deport asylum seekers to Rwanda. That faces a critical test on Wednesday when the country’s Supreme Court is scheduled to rule on its legality following a series of challenges.Mr. Sunak visiting an Ikea distribution center in Dartford, Kent, in June.Pool photo by Jack HillThe decision to bring back Mr. Cameron, who led the Conservatives between 2005-16, seemed at odds with Mr. Sunak’s recent claims at his party’s annual conference to be an agent of change.It also underscored a constitutional requirement of Britain’s political system that ministers hold a seat in Parliament so they can propose legislation and be held to account by fellow lawmakers. As a consequence Mr. Sunak on Monday nominated Mr. Cameron for a seat in the House of Lords, Parliament’s less powerful, unelected upper chamber.It is not the first time in the modern era that a foreign secretary has been a member of the House of Lords, rather than the House of Commons: Peter Carington, who became Lord Carrington — and as such, gained a second r in his name — filled that role between 1979-82. He resigned amid the Falklands crisis, when troops from Argentina occupied a British overseas territory in the South Atlantic in 1982, sparking a brief conflict.While the situation is not unique, Mr. Cameron’s status as a member of the House of Lords has already raised tensions among lawmakers in the House of Commons as he will normally speak not to them, but to an assembly of unelected members of the upper chamber.Lindsay Hoyle, speaker of the House of Commons, said on Monday that he was looking into ways in which the new foreign secretary could be held accountable by elected lawmakers. It was “especially important” that they should be able to scrutinize his work, “given the gravity of the current international situation,” Mr. Hoyle said in Parliament. More