  Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an
integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following
generalization of the Erd\H{o}s-Falconer distance problem: If $|A|\gg
q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then we have
\[\Delta_{k}(A)=\{|x_1+\cdots+x_k|\colon x_i\in A\}\supseteq \mathbb{F}_q^*.\]
In this paper, we provide improvements and extensions of their result.
