ROCK HILL, S.C. — Addressing a conference of African-American church congregations in this vote-rich city, Pete Buttigieg quoted scripture on Sunday morning and extolled his “Douglass Plan” to combat racial inequities in America, one of several attempts this weekend to confront his strikingly low support among black voters.

But Mr. Buttigieg also undertook a delicate task before the African Methodist Episcopal worshipers. As a gay, married man addressing a denomination that does not allow same-sex marriage rites, he tried to seek common ground over being members of minority groups whose civil rights have come under attack. It was a nod to his sexuality, following the disclosure last week that the Buttigieg campaign held focus groups that found some black voters in South Carolina were uncomfortable with a gay man as president.

“All of us in different ways have been led to question whether we belong,” Mr. Buttigieg told the pews of black worshipers. “And I know what it is to look on the news and see your rights up for debate. All of us must extend a hand to one another. Because I also know what it is to find acceptance where you least expect it.”

As Mr. Buttigieg increasingly presents himself to Democrats as a younger, moderate alternative to former Vice President Joseph R. Biden Jr., he is struggling badly to compete against one of Mr. Biden’s strengths: deep connections to black voters. Nowhere is that problem greater than in South Carolina, which votes fourth in the Democratic nomination fight in February and is the first state where black voters are decisive — a critical test that could be a prologue for primaries in March where African-Americans will also be influential.