Has New York Hit a Progressive Plateau? The Mayor’s Race Is a Key Test.

“For the other candidates,” he added, “that really cedes that discussion to Adams.”

An NY1-Ipsos poll released on Monday found that 46 percent of likely Democratic voters viewed crime and public safety as the top priority for the next mayor. A staggering 72 percent said they somewhat or strongly agreed that the Police Department should put more officers on the street.

A quarter of likely voters polled for the survey identified themselves as more progressive than the Democratic Party. Nearly an equal share, 22 percent, said they were more centrist or conservative. Just over half called themselves “generally in line with the Democratic Party,” which has shifted significantly to the left as a whole in recent years.

Whatever the primary results, party strategists warn against drawing sweeping conclusions from a post-pandemic Democratic municipal contest that is likely to be a low-turnout affair.

Still, city elections in recent years have been important barometers of grass-roots energy, including the 2019 race for Queens district attorney, where Tiffany L. Cabán, who ran as a Democratic Socialist, nearly defeated Melinda Katz, a veteran of New York politics.

In this year’s race for Manhattan district attorney, at least three contenders have sought to emulate Ms. Cabán. But the three — Tahanie Aboushi, Eliza Orlins and Dan Quart — have struggled to win support. A more moderate candidate, Tali Farhadian Weinstein, has led in fund-raising, including $8.2 million in contributions that she recently made to her own campaign, and the few available polls.

Tensions on the left burst into public view when Zephyr Teachout, a candidate for governor in 2014, argued on Twitter that Mr. Quart, Ms. Orlins and Ms. Aboushi had no path to victory.

That drew a sharp response from Cynthia Nixon, who challenged Gov. Andrew M. Cuomo from the left in the 2018 primary and supports Ms. Aboushi. (Ms. Teachout supports Alvin Bragg, a former prosecutor who has also won the backing of progressive groups.)