WASHINGTON — Faith and politics intersected on Sunday when discussions of how Donald J. Trump handled a question about President Obama’s religion evolved into an awkward debate over whether a Muslim could ever be commander in chief.

Several Republican presidential candidates offered halting responses to the question when it was posed to them on Sunday talk shows, highlighting the difficulty the party faces in attracting new constituencies while appealing to its core supporters. The dynamic also underscores the challenge Mr. Trump has brought to the campaign by igniting controversies that other candidates must address.

The issue was stirred last week when Mr. Trump did not correct an attendee at a rally in New Hampshire who asserted that Mr. Obama is a Muslim.

In response to criticism in the subsequent days, Mr. Trump, who spearheaded the “birther” movement to investigate Mr. Obama’s nationality, said it was not his job to defend the president.