Asked about a federal takeover of Rikers, Ms. Clark said that she was “in favor of anything that is going to bring justice, that’s going to make Rikers Island more humane and more safe, but it’s not my decision.”

Ms. Clark is leading the race by some traditional campaign markers. She has more money on hand — $281,000 according to a report filed on June 16, compared with just under $16,000 for Ms. Cohen — and the backing of numerous unions and Democratic Party heavyweights, including Senator Chuck Schumer, Attorney General Letitia James, Bronx Borough President Vanessa Gibson, and Assembly Speaker Carl E. Heastie.

“Darcel is a strong candidate, she’s been a lifelong Bronxite, she knows intimately what the issues of the Bronx are, and I just don’t see the challenger bringing that to the table,” said Virginia Krompinger, president of the Benjamin Franklin Reform Democratic Club, which endorsed the incumbent.

Ms. Cohen has won the support of voters and organizations explicitly looking for a change — including a number of formerly incarcerated people who were exonerated. Amanda Litman, the co-executive director of Run for Something, a progressive group that recruits political candidates, said her group had endorsed Ms. Cohen because “she knows the system in and out, she has a really strong progressive vision for what the office can be and what the office can do.”

Turnout in New York City’s primary elections is not expected to be high — and it remains to be seen how focused voters are on the district attorney contest in the Bronx.

Ayisha Khalid, a college student studying politics and criminal justice, answered the door when Ms. Cohen knocked, listened to her pitch and appeared to appreciate the candidate’s ideas about providing second chances for people who commit crimes. Still, she said, “I have to read more about it, because I had no clue.”