Reducibility of $rx^m+p^e f(x)$ for Large Primes $p$

We study the reducibility over $\mathbb{Q}$ of $F_{p,e}(x)=rx^m+p^ef(x)$, where $r\in\mathbb{Z}\setminus\{0\}$, $f\in\mathbb{Z}[x]$, and $0\le m<n:=\mathrm{deg}\, f$, for primes $p$ above explicit coefficient-dependent thresholds. For arbitrary $e\ge1$, we determine the degrees, endpoint $p$-adic va