The residue-counts of $x^2+a\mathrm{/}x$ modulo a prime

For a prime $p>3$ and $a\in \mathbb{Z}$ with $p\nmid a$ let $V_p(x^2+\frac{a}{x})$ be the residue-counts of $x^2+\frac{a}{x}$ modulo $p$ as $x$ runs over $1,2,\dots ,p-1$. In this paper, we obtain an explicit formula for $V_p(x^2+\frac{a}{x})$, which is concerned with cubic residues and binary quadratic forms.