We know that Weil's bound tells $$\left| \sum_{x \in \mathbb{F}_{q}} \chi(f(x)) \right| \leq (d-1)\sqrt{q},$$ where $\chi$ is a multiplicative character of order $m$, $f$ is not an $m$th power in $\mathbb{F}_{q}[x]$, and $d$ counts the number of distinct roots of $f$ in $\overline{\mathbb{F}_{q}}$.
I am curious about the following sum $$\left| \sum_{x_1 \in \mathbb{F}_{q}}\cdots \sum_{x_n \in \mathbb{F}_{q}} \chi(f(x_1,\ldots,x_n)) \right|,$$ where again $\chi$ is a multiplicative character of order $m$ and $f$ is not an $m$th power in $\mathbb{F}_{q}[x_1,\ldots,x_n]$. Could anyone give me some references on the bound of this sum?