In Silas Mariner does Godfrey have any right to try to win Nancy's approval?
This is a difficult question. Godfrey absent-mindedly (to put it nicely) married Molly and then abandoned her and the child on account of Molly's addiction and Godfrey's infatuation with Nancy. So, he is flawed. Godfrey is compared to Dunstan since they are brothers and in that respect, the reader finds it at least possible to sympathize with Godfrey because he is the more moral man; he just lacks the backbone to live up to his mistakes (until the end, when it is too late).
One of the themes in the novel is that everyone gets what they deserve and if you follow that, then, in the grand scheme of things, Godfrey’s marriage to Nancy is consolation for the ironic fact that they cannot bear children and he can never publicly acknowledge Eppie as his daughter: and having a child is the sole thing he’s missing in his life. Since, he abandoned his first wife and daughter, karma comes back to haunt him in this way. Likewise, Dunstan gets what he deserves and Silas gets Eppie.
If you’re asking if Godfrey deserves Nancy’s hand in marriage, it is hard to say. Godfrey was born into the upper strata of society in this town and it is expected of him to marry her. Luckily for him, he wants to marry her. Because he is born into this position in society, he hid the fact of his first wife and child as it would be a disgrace. So, you can either sympathize with the social pressure he faced or conclude that he doesn’t deserve Nancy’s hand in marriage because his mistakes outweigh the pressure he faces from society and his father. To go a bit further, you could make the argument that any one of the more hard-working, upstanding (and never mentioned) members of the working class are more deserving in a kind of Cinderella story format. In the end, Nancy would only have someone of Godfrey’s standing.
If you were asking if Godfrey was right to seek Nancy’s approval for the mistakes he’s made (hiding his wife and child), again, that’s up to you, but Nancy says he’s worth it after giving her 15 years of a happy marriage.