Last year, Prime Minister Rishi Sunak said in a speech that he was proud to be Britain’s first prime minister of Asian heritage, but “even prouder that it’s just not a big deal.”

On Friday, Mr. Sunak said he was “hurt” and “angry” after a man campaigning on behalf of Reform U.K., an anti-immigration party, was recorded on video using a racist slur to describe him. The same man also called for migrants to be used as target practice.

The comments appeared in an exposé by Channel 4 News, in which an undercover investigator secretly filmed Reform campaigners in Clacton, a seaside area north east of London. The party’s leader, the veteran political disrupter Nigel Farage, hopes to win his first parliamentary seat there.

The investigation, broadcast on Thursday night, raised uncomfortable questions about Reform, which has shaken up the country’s general election campaign ever since Mr. Farage reversed an earlier decision not to stand for Parliament.

In the weeks since, the insurgent party has risen in the polls, at one point threatening to overtake Mr. Sunak’s Conservatives as the second-most-popular party, before recently falling back. But it has also come under fierce criticism after a number of its candidates were found to have made incendiary statements.

Mr. Farage initially said he was “dismayed” by the comments broadcast in the Channel 4 News investigation, adding, “Some of the language used was reprehensible.”

But on Friday, after it emerged that the man at the center of the furor, Andrew Parker, was a part-time actor, Mr. Farage claimed that his party had been the victim of “a total setup,” an allegation that Channel 4 News strongly rejected.

The investigation also recorded homophobic comments being made by George Jones, an activist closely linked to Reform U.K.

Mr. Jones, an aide who has also worked for two other parties formerly led by Mr. Farage, was heard describing a Pride emblem on a police car as a “degenerate flag” adding that, should Reform U.K. form a future government, “our police officers will be paramilitaries,” and suggesting that the party should “bring back the noose.”

The comments broadcast in the TV exposé prompted anger from lawmakers across the political spectrum. The strongest condemnation, however, was for Mr. Parker, who described Islam as “the most disgusting cult out,” suggested army recruits should carry out “target practice” by shooting at migrants arriving on the British coastline and used a racial slur to describe the prime minister.

On Friday, Mr. Sunak told broadcasters that “it hurts and it makes me angry” that his two daughters “have to see and hear Reform people who campaigned for Nigel Farage” using such offensive language directed at their father.

The prime minister repeated the slur himself as he criticized it, saying: “I don’t repeat those words lightly. I do so deliberately because this is too important not to call out clearly for what it is.”

Mr. Farage has questions to answer, Mr. Sunak added, saying that such corrosive and divisive behavior “tells you something about the culture within the Reform party.”

In a statement to Channel 4 News, Mr. Parker said that “neither Nigel Farage personally or the Reform Party are aware of my personal views on immigration,” adding that he would “like to apologize profusely to Nigel Farage and the Reform Party if my personal views have reflected badly on them and brought them into disrepute as this was not my intention.”

Channel 4 News said in a statement that it “did not pay the Reform U.K. canvasser or anyone else in this report. Mr. Parker was not known to Channel 4 News and was filmed covertly via the undercover operation.”

The broadcaster added, “We strongly stand by our rigorous and duly impartial journalism which speaks for itself.”

The TV exposé is the second significant setback for Mr. Farage, who stunned the Conservative Party when he unexpectedly announced this month that he would take over the leadership of Reform and run in the general election.

While Reform is unlikely to win many seats under a British electoral system that favors the two largest parties, it could poach enough votes from the Tories to ruin their chances in a significant number of seats, compounding their woes in an election that polls predict they will lose.

Until recently, the Conservatives had been reluctant to criticize Mr. Farage and his party directly, in part because they hope to woo some voters who sympathize with Reform.

But even before the latest outcry, Reform’s surge in the polls appeared to have slipped back somewhat after Mr. Farage said in a TV interview that the West had provoked the invasion of Ukraine by Russia’s president, Vladimir V. Putin. That is not a popular position in Britain, where support for the Ukrainian government remains strong and there is little affection for Mr. Putin.

Some Reform election candidates had already come under scrutiny for past comments, with one saying that Britain should have remained neutral in the fight against the Nazis, and another using antisemitic tropes and claiming that Jewish groups were “agitating for the mass import into England of Muslims.”

The party has blamed some of its problems on growing pains and has threatened to take legal action against a private company it paid to vet candidates.