Absolutely, repenting and redemption are highly praised by Hawthorne. Dimmesdale is the best example of this. He suffers throughout the whole story, getting weaker and weaker because his guilt is eating away at him. When he confesses upon the scaffold at the end, he is able to die at peace. He receives a blessing from Pearl, who recongizes that he has now repented for his sin:
Pearl kissed his lips. A spell was broken. The great scene of grief, in which the wild infant bore a part, had developed all her sympathies; and as her tears fell upon her father's cheek, they were the pledge that she would grow up amid human joy and sorrow, nor for ever do battle with the world, but be a woman in it. Towards her mother, too, Pearl's errand as a messenger of anguish was all fulfilled.
Dimmesdale tells Hester that God is forgiving and merciful, and that it was his duty to come forward to the people and be honest about his sin. Having done so, he can die and ascend to heaven.
Hester is, of course, another example. She repents throughout her life, and is rewarded not only by everlasting forgiveness, but by the townspeople. Hawthorne makes clear that the people of the town altered their opinion of her, that her "A" was transformed from adultery to able. Thus, she has been forgiven for her sins.
You present two questions, though - you ask if the sin can be purged from a person's soul. That seems different than repentance. This sin, according to Hawthorne, stays with the person, as Hester's A stayed with her. However, they can earn forgiveness for their sin.