JACKSON, Miss. — When Mike Espy, the Democrat challenging Senator Cindy Hyde-Smith of Mississippi, faced his opponent at a debate ahead of this Tuesday’s runoff election, he had to make a choice: confront Ms. Hyde-Smith over her comments about attending “a public hanging,” which evoked the state’s racist history, or take a milder approach to avoid alienating the conservative-leaning white voters who will most likely decide the election.

He chose the latter.

“The world knows what she said, the world knows that those comments were harmful and hurtful,” Mr. Espy said afterward, sounding not entirely convinced.

In a state where politics has long been cleaved by race, Mr. Espy was reckoning with a conundrum that Democrats face across the South — from Mississippi and Alabama, which have been hostile to the party for years, to states like Florida and Georgia that are more hospitable in cities but still challenging in many predominantly white areas. Even as they made gains in the 2018 elections in the suburbs that were once Republican pillars, Democrats are seeing their already weak standing in rural America erode even further.

Now, as Democrats mount a last-minute and decidedly against-the-odds campaign to snatch a Senate seat in this most unlikely of states, they are facing the same problem that undermined some of their most-heralded candidates earlier this month.