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 ≥ 7 q \geq 7 , every residue class modulo an irreducible polynomial F ∈ F q [ X ] F \in \mathbb {F}_q[X] has a non-constant, square-free representative which has no irreducible factors of degree exceeding deg ⁡ F − 1 \deg F -1 . We also give applications to generating sequences of irreducible polynomials.

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We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear sums and to some questions regarding smooth and square-free polynomials in residue classes.