Policing problems in South Bend came to national attention on June 16, when a white sergeant fatally shot a 54-year-old black resident, Eric Logan. The officer’s body camera was not turned on, which was widely seen as a sign of lax standards in the department. Mr. Buttigieg found himself flying home again, regularly, to face the fury of some black citizens and the frustrations of many others.

It is the great paradox of Mr. Buttigieg’s presidential candidacy: His record on public safety and policing, once largely a footnote in his political biography, has overshadowed his economic record in South Bend, which he had spent years developing as a calling card for higher office.

“When he came in, the goal was to help turn the city around. That had nothing to do with the police department,” said Kareemah Fowler, until recently the South Bend city clerk.

Mr. Buttigieg’s image as a young, results-oriented executive continues to make him popular with many upper-income white liberals. They have delivered an overflowing war chest to his campaign: He had the best recent fund-raising quarter of any Democrat in the race, pulling in $24.8 million.

But criticism of Mr. Buttigieg’s oversight of the police has damaged his viability as a Democratic presidential candidate, given the huge influence of black voters in choosing the party’s nominee. He has slipped in the polls in recent months, from double-digit poll numbers in Iowa and New Hampshire in the spring to the single digits more recently. In a recent Fox News poll, he earned less than 1 percent support from black Democratic primary voters.