Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial and let $p$ be a prime number. I'm looking for results about $$N_f(p^k) := \#\{(f(n) \bmod p^k) : n \in \mathbb{Z}\},$$ as $k \to +\infty$, where $(a \bmod b)$ is the remainder of $a$ divided by $b$.
If $N_f(p) = p$ and $f^\prime(x) \not\equiv 0\bmod p$, for all $x \in \mathbb{Z}$, then using Hensel's lemma it can be proved that $N_f(p^k) = p^k$ for all $k \geq 1$. However, I did not find more general result.
Thank you for any suggestion.