Since the election of Ronald Reagan, white Catholics have flocked to Republican nominees for a raft of reasons, including their stances on social issues like abortion and same-sex marriage.

But this year, something seems different.

“Trump is the exception to the rule,” Carol Robinson, 67, said as she left an afternoon prayer meeting in this a Philadelphia suburb with other enthusiastic supporters of Hillary Clinton. “He’s a loose cannon.”

Roman Catholics are the country’s second-largest religious group after evangelical Protestants, and they are as diverse as the country itself, with young liberals, cultural conservatives and, increasingly, Democratic-leaning Hispanics.

But now, the Clinton campaign senses a rare opportunity to block Mr. Trump’s narrow path to victory by making inroads with a core part of the church: white Catholics, a prized group of voters who have defied predictions this year.