Joe Darby, a prominent pastor in Charleston, S.C., was discussing the Democratic presidential field with fellow clergymen when Pete Buttigieg’s name came up. A fellow pastor quickly interjected. “Isn’t that the dude who kissed his husband on TV?” the person asked skeptically, according to Darby. The exchange highlights a major obstacle for Buttigieg, who’s vaulted into the top tier of Democratic candidates without gaining traction among African Americans, according to recent surveys of national and South Carolina Democrats. But as the mayor of South Bend, Ind., devotes more effort to campaigning for black votes in the South and elsewhere, he...