[Here’s what’s coming up next on the primary calendar.]

Separately, each of the 14 candidates has their own constituencies to impress, and they face varying degrees of viability come November. But together, the group is a cross-cultural fraternity united by a common thread: America’s fraught racial history. There’s the tiptoeing around certain topics with an overt racial element, several said. Questions about their name, background or appearance can be as common as ones regarding policy issues. There’s also concern about failing, a belief that any mistake from a minority candidate would be punished in a way that’s far worse than one from their white opponents.

“We have to be excellent,” said Ms. Underwood, who served as a senior adviser at the Department of Health and Human Services before returning to her Illinois hometown to run against Representative Randy Hultgren, a four-term incumbent. “I recognize that we’re a trailblazer and we’re doing this historic thing, and we have to do it in a way that allows people to come behind us.”

Minority candidates “have to cross more T’s and dot more I’s,” said Colin Allred. “I know that.” Mr. Allred is a black 35-year-old former voting rights attorney and N.F.L. player who is running in Texas’ 32nd District, another mostly white community on the Democrats’ target list. He will face Representative Pete Sessions, a Republican who first assumed office in 1997.

The makeup of Congress skews disproportionately white, and among the minority House members, a large percentage represent gerrymandered districts made up predominantly of minority voters. (There are some exceptions, including black women on both sides of the aisle in Congress: Mia Love of Utah, a Republican whose district is 74 percent white, and Delaware’s Lisa Blunt Rochester, a Democrat whose district is 62 percent white.)

During an interview in Mr. Allred’s suburban Dallas hometown — a district that is 48 percent white and 26 percent Hispanic — he said he has sidestepped some issues on the campaign trail in order to avoid being deemed “too black.”

“There’s some topics I have to talk about differently, and there’s some subjects where I’m probably not going to be the one to lead on the issue,” Mr. Allred said, declining to be more specific. “You have to be honest and say, ‘You know what, as a young black man I’m probably not going to be the flag-bearer for that.’”