The two contenders to become Britain’s next prime minister clashed angrily over tax, immigration and health policy on Tuesday in a televised debate that at times descended into ill-tempered exchanges as the political rivals talked over each other.

The confrontation came exactly a month before a pivotal general election that will determine whether the opposition Labour Party can capitalize on its strong lead in opinion polls and end 14 turbulent years of Conservative-led government during which the party has had five different prime ministers.

Almost as soon as the debate started, Prime Minister Rishi Sunak claimed that his opponent, the Labour leader Keir Starmer, would raise taxes on Britons by 2,000 pounds, about $2,550, a year if he won the election, repeating the claim numerous times. “Absolute garbage,” Mr. Starmer eventually responded.

The Labour Party said that the figure was based on faulty assumptions, and Jonathan Ashworth, a senior party lawmaker, said in an interview with Sky News after the debate that Mr. Sunak was lying. But Mr. Starmer’s failure to clearly reject the claim early in the broadcast set the tone for what followed: a solid but defensive performance by the opposition leader against an energized and at times ruthless opponent.

One snap opinion poll of viewers declared Mr. Sunak a narrow victor, although Mr. Starmer was seen as more likable and more trustworthy. While the debate is unlikely to swing significant numbers of votes, Mr. Sunak’s performance may have steadied some nerves inside his anxious party.

With the Conservatives trailing badly in the opinion polls for more than 18 months, the broadcast was a chance for Mr. Sunak to revive his stalling campaign. After a gaffe-prone start, the prime minister’s prospects had seemingly worsened on Monday when Nigel Farage, a right-wing insurgent, made a surprise decision to run in the election.

For Mr. Starmer, the main objective was to avoid losing momentum ahead of a general election on July 4 that opinion polls say he is on course to win, perhaps comfortably.

There was no knockout blow in Tuesday’s hourlong debate, which was filmed in front of a studio audience in Salford, near Manchester, and was the first of two scheduled televised contests between Mr. Sunak and Mr. Starmer.

Animated but at times hectoring, Mr. Sunak was more aggressive in pushing his point, accusing Labour of having no plans for government and often talking over Mr. Starmer, despite pleas for calm from Julie Etchingham, the moderator.

But Mr. Sunak struggled to defend the Conservative Party’s 14-year governing record, and Mr. Starmer ridiculed his failure to cut waiting lists for treatment of more than seven million procedures in the health care system as he had promised.

“There were 7.2 million, there are now 7.5 million. He says they are coming down — and this is the guy who says he’s good at maths,” Mr. Starmer said of the prime minister.

“They are coming down from where they were when they were higher,” Mr. Sunak replied, prompting a burst of laughter from the audience.

In a familiar exchange of claims and counterclaims, Mr. Starmer said that the government had “lost control” of the economy, adding that it was ordinary people “who are paying the price.” Mr. Sunak argued that his plans were helping to revive economic growth and said that progress would be put at risk by Labour.

Televised debates for general elections are a relatively recent phenomenon in Britain, with the first taking place in 2010. The onus this time had been on Mr. Sunak to make an impact, in a broadcast that was described as “one of the last opportunities the prime minister has to change his party’s political fortunes,” by Lee Cain, who worked in Downing Street for Boris Johnson, one of Mr. Sunak’s predecessors.

Earlier on Tuesday, Mr. Farage, who has taken over as the leader of Reform U.K., a small hard-right party that campaigns to cut immigration, addressed a crowd of several hundred people in Clacton-on-Sea, which is part of the area he plans to contest in the general election.

Playing on his reputation as a political disrupter, Mr. Farage appealed to voters to send him to Parliament “to be a bloody nuisance.” Not all bystanders were friendly, however, and one protester threw what appeared to be a large milkshake over him. A woman was later arrested.

A leading proponent of Brexit, Mr. Farage has tried and failed seven times to become a member of the British Parliament. But analysts believe he has a fair chance this time in Clacton, an area that voted strongly for Britain’s exit from the European Union, and that was once represented by a lawmaker for the U.K. Independence Party, the pro-Brexit party Mr. Farage used to lead.

Nationally, Reform U.K. is unlikely to win more than a handful of seats under the British electoral system, which favors the two largest parties and makes it very hard for small parties to break through.

But Mr. Farage’s party tends to take more votes from the Conservatives than it does from Labour and could siphon thousands of votes that Mr. Sunak’s party won in the 2019 general election, potentially costing it dozens of seats.

Mr. Sunak on Tuesday made a new attempt to appeal to potential Reform voters, pledging to limit immigration by placing an annual cap on entrants.

Under his plans, an expert committee would recommend a maximum number of immigrants that would be allowed each year, and that would then be voted on by Parliament.

Labour dismissed the promise as meaningless, noting that previous Conservative election pledges to limit immigration had not been honored and that net migration had increased about threefold since the last election, in 2019.

At one point during Tuesday’s debate, Mr. Sunak accused Labour of having no plan to curb the number of asylum seekers crossing the English Channel on small boats. And he hinted that he would be willing to take Britain out of international agreements if he were to remain prime minister and was thwarted in his efforts to put some of those arriving on the British coast on one-way flights to Rwanda.

Mr. Starmer described that scheme as an “expensive gimmick,” and attacked Mr. Sunak over the surge in legal immigration since the 2019 general election. “The prime minister says ‘It’s too high,’” Mr. Starmer said, adding, “Who’s in charge?”