How does Reverend Arthur Dimmesdale intercede on behalf of Hester? (Chapters 6-11)
In chapter 8, when Hester stands to lose Pearl, she looks to Dimmesdale to intervene on her behalf and convince the Governor that Pearl should remain with her. Dimmesdale does just this. He argues that Pearl may serve to save Hester's soul, which is otherwise lost to Satan.
As far as the other chapters, Dimmesdale doesn't really help Hester directly. He does try to publicly state his sin, but he only ends up saying that he is a sinner and a liar. During these chapters, he does not confess his part in the adultery, but his own letter A does become embedded on his chest.
In Chapter 8, Governor Bellingham and the authorities interview Pearl to see if she is being raised properly. They are considering removing Pearl from Hester's care in order to save Pearl's soul. Dimmesdale intervenes with the authorities after Hester demands that he help. He posits that if Hester loses Pearl, then Hester's soul might be lost, so Pearl should be left with Hester to save Hester's soul. A compromise is reached whereby Hester keeps Pearl, but Pearl is supervised at church.