Suppose that $P,Q\in\mathbb F_p[x]$ are monic, nonconstant polynomials with $\deg P+\deg Q=p$. Let $R$ be the remainder of division of $PQ$ by $x^p-x$; that is, $$PQ=(x^p-x)+R(x),\quad \deg R<p.$$ Can one express explicitly the degree of $R$ in terms of the degrees $m=\deg P$ and $n=\deg Q$, or at least say something strong about $\deg R$?
In addition to $m+n=p$, one can assume that $p$ is large, $m<0.9p$, and also $\deg R<0.9p$.
Assume $m,n < .9p$.
Here's a partial result . . .
Claim:$\;$For any integer $r$ in the range $.1p < r < .9p$, it's always possible to have $\deg R = r$.
Proof:$\;$Fix an integer $r$ such that $.1p < r < .9p$, and let $P,Q$ be given by \begin{align*} P&=\prod_{k=0}^{r-1}(x-k)\\[4pt] Q&=1+\prod_{k=r}^p(x-k)\\[4pt] \end{align*} Then we have $m,n < .9p$ and $m+n=p$, and from the identity $$x^p-x=\prod_{k=0}^{p}(x-k)$$ it follows that $$PQ=(x^p-x)+P$$ hence $R=P$, so $\deg(R)=r$.
Note:$\;$If instead of the restriction $m,n < .9p$, we only require $m,n < p$, then using the same construction, we get that $r$ can be any integer in the range $1 \le r \le p-1$.