Final answer:
The Republicans revealed that Grover Cleveland had fathered a child out of wedlock, but he still won the election due to his reformist reputation.
Step-by-step explanation:
The story that Republicans broke on Grover Cleveland during his presidential campaign was the personal charge of him fathering a child out of wedlock. Despite this scandal, Cleveland's reputation as a reformer and attacker of machine politics in New York City helped him secure a narrow victory. His electoral success was attributed in part to winning the Mugwump vote, which was crucial during a time when the Democratic Party aimed to return to the White House after a nearly thirty-year absence.