Money is a serious matter in a relationship, it is not wise to ask your girlfriend for financial support and I do not think there is a right way to ask for such help. Money has shown to be the source of many problems in a relationship.
I will suggest that you work with the little that you have, share your ideas with her. Let her notice your effort and be willing to help you financially; this will make your girlfriend respect you and not think that you want to take advantage of her. As far as finances are concerned, it is best that is voluntarily given.
You should not look to a girl friend or anyone except family or employers for financial support. The exception would be if you have a long-standing relationship with a girl, say well over a year, and that relationship involves both of you helping the other out.
If you except a grildfriend to financially support you, what does that say about you as a person? I'm afraid the answer to that question is very negative. Think whether your male mates would respect you, or th girls' family...or anyone else, for that matter. If you need more income, take a part-time job and earn it. You'll earn the respect of others as a result, too.