While Georgia’s House special election Tuesday is a test of whether Republicans will cross party lines to support a Democrat, another contest the same day in South Carolina aims to measure a different proposition: Can a white Democrat drive enough black voter turnout to score an upset.

The race is to replace former Rep. Mick Mulvaney, who President Donald Trump tapped to serve as his budget director. Archie Parnell, a former Goldman Sachs executive, is running on the Democratic ticket against Republican Ralph Norman, who narrowly...