Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. I would like to have an estimate for the number of representations $R(n)$ of $n \in \mathbb{Z}$ in the form $$ f(x) - f(y) = n, \qquad x,y \in \mathbb{N}. $$ If $f(x)=x^2$, then the number of representations is a classical problem, and $R(n) = \mathcal{O} (|n|^\varepsilon)$. Is the same true when $f(x)$ is an other polynomial of degree 2 or higher?
(If you have an answer, please include a reference or good argument. Thanks.)