The trick for Republicans and their allied outside groups is figuring out how to avoid conspicuously embracing the president without alienating conservative voters who would view any overt rebuff as a betrayal.

“That is the question we are trying to answer right now,” said Ralph Reed, whose Faith and Freedom Coalition is based in Georgia and is involved in the special election.

As for Mr. Trump, Mr. Reed added: “I don’t think you really look to broadcast him. You narrowcast him.”

The Faith and Freedom Coalition’s “narrowcast” comes in the form of about 30,000 one-page leaflets that volunteers will slip into church bulletins across the district ahead of the election. They list a series of issues that are resonant with conservatives and at the top of Mr. Trump’s agenda — like Planned Parenthood funding, “abortion on demand,” freedom of speech for churches and “amnesty for illegal immigrants” — and offer a side-by-side comparison of Ms. Handel’s and Mr. Ossoff’s stands.