Thank you both! Are you perhaps aware of a general result that bounds the number of lattice points inside a convex region in terms of successive minima? I know of Davenport's theorem (from the replies to related questions), but it does not seem to give sharp bounds for what I need.

Actually, I had in mind the general case, not necessarily prime powers. But of course, if something can be said in certain special cases, that would also be interesting (apart from the case of prime $q$).

Yes, the case of prime $q$ is very easy. This looked to me like a problem that has been studied before, so I was hoping for a reference... Should the prime power case also be obvious?