  Given a monic polynomial $f(X)\in \mathbb{Z}_m[X]$ over a residue ring
$\mathbb{Z}_m$ modulo an integer $m\ge 2$ and a discrete interval $\mathcal{I}
= \{1, \ldots, H\}$ of $H \le m$ consecutive integers, considered as elements
of $\mathbb{Z}_m$, we obtain a new upper bound for the additive energy of the
set $f(\mathcal I)$, where $f(\mathcal I)$ denotes the image set $f(\mathcal I)
= \{f(u):~u \in \mathcal I\}$. We give an application of our bounds to
multiplicative character sums, improving some previous result of Shkredov and
Shparlinski~(2018).
