Square-free smooth polynomials in residue classes and generators of irreducible polynomials

Building upon the work of A. Booker and C. Pomerance [Proc. Amer. Math. Soc. 145 (2017), pp. 5035–5042], we prove that for a prime power q \geq 7, every residue class modulo an irreducible polynomial F \in \mathbb{F}_q[X] has a non-constant, square-free representative which has no irreducible factors of degree exceeding \deg F -1. We also give applications to generating sequences of irreducible polynomials.