Final answer:
Hester visits Governor Bellingham's mansion to deliver gloves and to address her concern about rumors that the authorities might take her daughter Pearl away from her.
Step-by-step explanation:
Hester is visiting Governor Bellingham’s mansion primarily to deliver a pair of gloves but also to address a personal concern. She is worried about the rumors that the authorities are considering taking her daughter, Pearl, away from her. This concern is due to the unconventional circumstances of Pearl's birth and Hester’s own public shaming for adultery. Hester believes that Pearl being taken from her would be an unjust and devastating event, so she goes to plead her case directly to Governor Bellingham.