I know that Governor Bellingham is a minor character, but what impact does he have on the plot of the book?"The Scarlet Letter" by Nathaniel Hawthorne

With a pair of gloves, fringed and embroidered to order, Hester and Pearl approach the mansion of Governor Bellingham.  This mansion is impressive:

...there was the freshness of the passing year on its exterior, and the cheerfulness, gleaming forth from the sunny windows, of a human habitation, into which death had never entered.  It had, indeed, a very cheery aspect; the walls being overspread with a kind of stucco, in which fragments of broken glass were plentifully intermixed; so that, when the sunshine fell a slant-wise over the front of the edifice, it glittered and sparkled as if diamonds had been flung against it by the double handful.  The brilliancy might have befitted Aladdin's palace, rather than the mansion of a grave old Puritan ruler.

As they approach the door, Hester and Pearl encounter the "serving-man," an indentured servant of the governor's.  This enslaved man and the opulence of the governor's mansion belie the Puritan simplicity.  Then, on the "Elizabethan age" elaborately carved table rests a tankard,

at the bottom of which had Hester or Pearl peeped into it,...have seen the frothy remnant of a recent draught of ale.

A "bright panoply of armor, an ancestral relic, is suspended at the center of the oaken panels.  When Governor Bellingham sees Pearl, he declares that he has not seen "the like" since his "days of vanity" in court in England when he was admitted to a masque.  This admission, along with the evidence of a draught of ale, indicate that the governor is hardly the ascetic Puritan.  Added to his Puritanical failings, his sister is a witch who invites Pearl to the black mass later on.

Again in contrast to the grey Puritan existence, in Chapter XXI the Governor reappears in elaborate garb for the New England Holiday.  His presence is a reminder that the residents "had not been born to an inheritance of Puritan gloom." Only a

dim reflection of a remembered splendor, a colorless and manifold diluted repetition of what they had beheld in proud old London...might be traced in the customs ....

Governor Bellingham's dress and mansion are reminders of what the Puritans gave up as well as serving notice of the hypocrisy of their austere religion that holds such as he--elaborateness and ale included--in high regard.  His involvement in the decision of whether Hester should keep her child serves to underscore the fact that for Puritans religion and the law are nearly identical.