  In this paper, we prove a new point-sphere incidence bound in vector spaces
over finite fields. More precisely, let $P$ be a set of points and $S$ be a set
of spheres in $\mathbb{F}_q^d$. Suppose that $|P|, |S|\le N$, we prove that the
number of incidences between $P$ and $S$ satisfies \[I(P, S)\le
N^2q^{-1}+q^{\frac{d-1}{2}}N,\] under some conditions on $d, q$, and radii.
This improves the known upper bound $N^2q^{-1}+q^{\frac{d}{2}}N$ in the
literature. As an application, we show that for $A\subset \mathbb{F}_q$ with
$q^{1/2}\ll |A|\ll q^{\frac{d^2+1}{2d^2}}$, one has \[\max \left\lbrace |A+A|,~
|dA^2|\right\rbrace \gg \frac{|A|^d}{q^{\frac{d-1}{2}}}.\] This improves
earlier results on this sum-product type problem over arbitrary finite fields.
