He had vowed to clean up Britain’s government after months of scandal and disarray under his predecessors. But on Friday, Prime Minister Rishi Sunak lost another top minister, as his deputy, Dominic Raab, resigned following an investigation that found he had bullied subordinates. Mr. Raab, one of Mr. Sunak’s most loyal political allies, had long denied allegations of abusive behavior. But the investigation, by an independent barrister, examined eight cases in which civil servants accused Mr. Raab, 49, who also served as justice secretary, of mistreating them. In at least one case, it described his behavior as “an abuse or misuse of power in a way that undermines or humiliates.” Mr. Raab is the third cabinet minister in six months to leave over ethics issues, illustrating the hurdles Mr. Sunak has faced in fulfilling his promise to lead a government of “integrity, professionalism and accountability at every level.” The drama over Mr. Raab’s fate played out over two days in Downing Street, suggesting that he initially dug in his heels about the report, which he claimed on Friday had largely exonerated him, and that Mr. Sunak was not without his own qualms about the investigation, which delved into allegations going back several years. In a warmly written letter accepting Mr. Raab’s resignation, Mr. Sunak noted, “You had — rightly — undertaken to resign if the report made any findings of bullying whatsoever. You have kept your word.” While Mr. Raab’s exit will help Mr. Sunak turn the page after a season of scandal, political analysts said it would have sent a stronger message if the prime minister had quickly fired his deputy, rather than allowing him to quit. “He’s going to be criticized for not making a decision straight away, for not cutting the rope and for not sacking Raab,” said Tim Bale, a professor of politics at Queen Mary University of London. “But I think in the longer term, all that’s important for most people is that an alleged bully has been disappeared.” Mr. Sunak rose to power after his former boss, Boris Johnson, was caught in a cascade of scandals, most prominently a series of Downing Street parties that violated lockdown rules. Liz Truss, who succeeded Mr. Johnson, was forced out after her trickle-down tax cuts backfired in financial markets. Mr. Sunak is credited with stabilizing Britain’s economic policy since he took over in October. But he has had a tougher time vanquishing a culture of questionable behavior that took root under Mr. Johnson. It has continued, partly because several ministers, Mr. Raab among them, were reappointed after serving in the Johnson government. The inquiry report, published Friday, found that when Mr. Raab was foreign secretary, he acted in a way that was “intimidating, in the sense of unreasonably and persistently aggressive conduct in the context of a work meeting.” At the Justice Ministry, the report said, Mr. Raab complained to officials about the absence of basic information, referred to employees’ “obstructiveness” and described some work as “utterly useless” and “woeful.” Still, the document concluded that the nature of those complaints at the Justice Ministry made it “unsuitable as a basis for any findings” about Mr. Raab’s conduct. And it found no clear evidence of bullying in several other cases. In his letter of resignation to the prime minister, Mr. Raab left little doubt that he was leaving his position reluctantly. The inquiry “dismissed all but two of the claims” against him, he argued, and the adverse findings about his behavior were flawed. “In setting the threshold for bullying so low, this inquiry has set a dangerous precedent,” he wrote. “It will encourage spurious complaints against ministers and have a chilling effect on those driving change on behalf of your government — and ultimately the British people.” Hours after Mr. Raab’s resignation, Oliver Dowden, secretary of state in the Cabinet Office, was named as the new deputy prime minister, and the job of justice secretary went to Alex Chalk, who had been a defense minister. The leader of the opposition Labour Party, Keir Starmer, said Mr. Sunak should never have appointed Mr. Raab and should have acted more swiftly to force him out. “What I think it shows is the continual weakness of the prime minister,” Mr. Starmer told Sky News. Mr. Sunak had already been criticized for not acting faster in dispatching two other cabinet ministers, Nadhim Zahawi and Gavin Williamson, for ethics issues. The uproar over Mr. Raab came just as Mr. Sunak had finally gained some political momentum, signing deals with the European Union on Northern Ireland and with France on migration issues. The perception that Mr. Sunak was a responsible leader had whittled down the Conservative Party’s large deficit in opinion polls, though Labour remains ahead by double digits. An ardent Brexiteer, Mr. Raab is one of a cadre of young politicians who rose to power in the bitter debate over whether Britain should leave the European Union. He was co-author of a book, “Britannia Unchained,” which described a vision of post-Brexit Britain as an agile, low tax, lightly regulated business. Other members of that group, including Kwasi Kwarteng, who served as chancellor of the Exchequer under Ms. Truss, have also been thrust out of power. Mr. Raab, who was promoted to the cabinet when Theresa May was prime minister, resigned his job as Brexit secretary in November 2018 in protest of her proposals for extracting Britain from the European Union. After an unsuccessful run for Conservative Party leader in 2019, he supported Mr. Johnson and was rewarded with the post of foreign secretary. He had a moment in the spotlight when Mr. Johnson became seriously ill with Covid and deputized Mr. Raab, as the ranking minister, to preside at cabinet meetings while Mr. Johnson was in the hospital. But he was later harshly criticized for staying on vacation on a Greek island during the chaotic withdrawal of British and American troops from Afghanistan. When Mr. Johnson was forced out as prime minister last July, Mr. Raab backed Mr. Sunak over Ms. Truss in the party leadership contest. After she defeated Mr. Sunak, she stocked her cabinet with loyalists and cast Mr. Raab into the political wilderness. His fortunes rebounded only weeks later when Ms. Truss was forced to resign, and Mr. Sunak finally captured 10 Downing Street, with Mr. Raab’s backing. He gave Mr. Raab, a lawyer, the post of justice minister and added the title of deputy prime minister, an unpaid, largely honorary position that does not come with an automatic right to succeed to prime minister. The New York Times