In Great Expectations, what financial request does Pip make of Miss Havisham and what does the request say about his character?

I think that the part you are asking about comes in Chapter 44.  It is right at the beginning of that chapter.

In that chapter, Pip asks Miss Havisham for money, but it is not for him.  Instead, he is asking her to arrange for Herbert Pocket to have some money without him knowing how it came about.  He wants Herbert to be able to succeed in business and he thinks that Herbert has been much nicer to Miss Havisham than the other relatives (so he doesn't deserve to be punished).

To me, this shows that Pip is a generous and unselfish person.