  For $ E\subset \mathbb{F}_q^d$, let $\Delta(E)$ denote the distance set
determined by pairs of points in $E$. By using additive energies of sets on a
paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset
\mathbb{F}_q^d $ are subsets with $|E||F|\gg q^{d+\frac{1}{3}}$ then
$|\Delta(E)+\Delta(F)|> q/2$. They also proved that the threshold
$q^{d+\frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an
improvement of this result in the unbalanced case, which is essentially sharp
in odd dimensions. The most important tool in our proofs is an optimal $L^2$
restriction theorem for the sphere of zero radius.
