$\newcommand\S{\mathcal S}$
Let $p$ be a prime, and suppose that $0.9(p-1)<q<p-1$. Suppose, furthermore, that $H,B,T\in\mathbb F_p[x]$ satisfy $\max\{\deg H,\deg T\}\le q-1$ and $\deg B\le p-1-q$. What can be said about $B$ given that the polynomial
  $$ \S(x) := x^p H(x) + x^q B(x) + T(x) $$
is fully reducible (splits completely into linear factors)? Is it true that for any $B$ there exist $H$ and $T$ such that $\S$ is fully reducible? What is the set of all those polynomials $B$ for which $H$ and $T$ can be found so that $\S$ is fully reducible? *Is there any reasonably strong condition that $B$ must satisfy in order for such $H$ and $T$ to exist?*