“Defund police, open borders, socialism — it’s killing us,” said Representative Vicente Gonzalez, a Democrat from South Texas who won just over 50 percent of the vote two years after he nearly captured 60 percent. “I had to fight to explain all that.”

The “average white person,” Mr. Gonzalez added, may associate socialism with Nordic countries, but to Asian and Hispanic migrants it recalls despotic “left-wing regimes.”

Representative Harley Rouda of California, an Orange County Democrat who narrowly lost his bid for re-election, said the party needed to deliver a more assertive and moderate message if it wanted to claim districts like his. Mr. Rouda, who is planning to run again in 2022, said he suffered with centrist voters and his district’s numerous Vietnamese-American voters, many of whom recoil from the rhetoric of socialism.

“This narrative that the Democratic Party is borderline socialist, we need to fight back harder on that because it’s simply not true,” he said. “We needed to be more forceful in defending the moderate position of the Democratic Party as a whole.”