  Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is large odd
prime power. In this paper, we improve some recent results on the additive
energy of the distance set, and on sumsets of the distance set due to
Shparlinski (2016). More precisely, we prove that for $\mathcal{E}\subseteq
\mathbb{F}_q^d$, if $d=2$ and $q^{1+\frac{1}{4k-1}}=o(|\mathcal{E}|)$ then we
have $|k\Delta_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q$; if $d\ge 3$ and
$q^{\frac{d}{2}+\frac{1}{2k}}=o(|\mathcal{E}|)$ then we have
$|k\Delta_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q,$ where
$k\Delta_{\mathbb{F}_q}(\mathcal{E}):=\Delta_{\mathbb{F}_q}(\mathcal{E})+\cdots+\Delta_{\mathbb{F}_q}(\mathcal{E})
~(\mbox{$k$ times}).$
