Number of irreducible polynomials whose compositions with monic monomials have large irreducible factors

Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor of degree $e$. Polynomials with these properties are important for justifying randomised algorithms for computing with matrix groups.