You certainly can say things like
Ask your tutor for homework . . .
Ask your tutor for more homework . . .
And this is probably the shortest, most direct way to ask for something, including homework.
As you have surmised, you may also ask in one of these ways:
Ask your tutor about homework . . .
Ask your tutor about more homework . . .
Ask your tutor about giving you some more homework . . .
Ask your tutor about the possibility of giving you some more homework . . .
What these last 3 versions are saying (without really saying it) is that your friend should be especially polite when she makes the request. So naturally, they are longer. You're communicating the instruction in a roundabout manner so that she will make the request in a roundabout manner.
Only you or your friend knows how much politeness is required by the situation.
As to your other suggestion, it is grammatically correct to say Then ask for some homework. It would be more idiomatically correct to say, Then ask for some homework. English strongly favors the use of articles and other determiners, especially when the noun in question is tangible or can be perceived that way—here, as a set of pages, perhaps.