What is Hester's real reason for visiting Governor Bellingham's home in chapter 7 of The Scarlet Letter?

Hester is indeed using the delivery of the gloves as a pretext to visit Governor Bellingham.  Her real reason for going to his home is to plead her case before him.  Hester has heard that there are some in the village who want to take her daughter Pearl away from her.  Amongst these people, Governor Bellingham has been one of the most vocal, and Hester wants a chance to argue on behalf of herself and of her child.

Governor Bellingham holds "an honorable and influential place among the colonial magistracy, and he has "much power and activity in the affairs of the settlement".  Hester has heard that "there (is) a design on the part of some of the leading inhabitants...to deprive her of her child...on the supposition that Pearl...(is) of demon origin".  Because of this, they are arguing that "a Christian interest in the mother's soul (requires) them to remove such a stumbling block from her path...(and) if the child, on the other hand, (is) really capable of moral and religious growth and possesse(s) the elements of ultimate salvation, then surely, it (will) enjoy all the fairer prospect of these advantages by being transferred to wiser and better guardianship than Hester Prynne's".  In her isolated state, Pearl is all Hester Prynne has, and she goes to the Governor's home seeking an interview with the esteemed gentleman so that she can plead her case before him and convince him to allow her to keep her child (Chapter 7).

In The Scarlet Letter, why does Hester visit the governor's house?

In Chapter VIII, Hester Prynne has been summoned to the Governor's Hall in order to confer with the state and church leaders who feel that Pearl should be taken from her mother in order to be properly raised. Upon arriving at the governor's mansion, Hester comes before the magistrate who is dressed in the elaborate ruff characteristic of King James's reign, a costume incongruous to the evidence that he lives in comfort and luxury. Likewise, the "old clergyman," the Reverend Mr. Wilson,

nurtured at the rich bosom fo the English Church, had a long established and legitimate taste for all good and comfortable things,

despite his public "reproof of such transgressions as that of Hester Prynne."

In her defiance of the condemnation befallen her, Hester dresses Pearl in a crimson dress that reminds the "good old Mr. Wilson" of the stained glass windows in the English churches of the "Papistry." Then, altering his demeanor to one of sternness, the Reverend Wilson informs Hester that they wish to remove the child to "other hands." This statement elicits a passionate response from Hester,

"God gave me the child....He gave her in requital of all things else which ye had taken from me.  She is my happiness! she is my torture, ...Pearl keeps me her in life! Pearl punishes me, too! Ye shall not take her! I will die first!"

Further, Hester appeals to the Reverend Dimmesdale to speak on her behalf, and he does so, contending that Pearl, the innocent child, keeps Hester from Satan since without her child, she may despair and fall prey to evil. Thus convinced by the young minister's plea, Mr. Wilson and the magistrate agreed to allow Hester to keep the incarnation of her sin. 

Why does Hester go to the Governor's Hall in The Scarlet Letter?

Hester goes to Governor Bellingham's mansion one day

with a pair of gloves which she had fringed and embroidered to his order, and which were to be worn on some great occasion of state.

She has been working as a seamstress, supporting herself and her little daughter, Pearl, with this work. Because of her great skill in producing beautiful items, she is often in demand for all kinds of garments and accessories. In many ways, the narrator suggests that this makes it seem as though the Puritans' vanity often outstripped their abhorrence of sin, as they coveted the Hester's productions regardless of her reputation. However, she is never asked to make anything having to do with a bride, as people seem to think that her adultery makes her an inappropriate person for this kind of work. When Hester goes to the governor's mansion, it is to deliver the pair of gloves which she has made for him and, presumably, collect her payment.