HMMT 2019 T9

Let $p>2$ be a prime number. $\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{p}$ for each nonnegative integer $k$. For example, $(x+2)(2 x+3)=2 x^{2}+2 x+1$ in $\mathbb{F}_{5}[x]$ because the corresponding coefficients are equal modulo 5 . Let $f, g \in \mathbb{F}_{p}[x]$. The pair $(f, g)$ is called compositional if

$$f(g(x)) \equiv x^{p^{2}}-x$$

in $\mathbb{F}_{p}[x]$. Find, with proof, the number of compositional pairs (in terms of $p$ ).