Sums of powers of subsets of an arbitrary finite field

We discuss the following problem: given an integer $n\geqslant 2$, a real number $\varepsilon\in (0,1)$, and an arbitrary subset $A\subseteq\mathbb{F}_q$ which is not contained in a multiplicative shift of a proper subfield of $\mathbb{F}_q$ and satisfies $|A|>q^{\frac{1}{n-\varepsilon}}$, where $\mathbb{F}_q$ is the finite field of $q=p^r$ elements, describe those positive integers $N$ and $m$ for which we have a set-theoretic equality $NA^m=\mathbb{F}_q$. In particular, we show that this equality holds for $m=2n-2$ and $N=N(n,r,\varepsilon)$.