Factorization type probabilities of polynomials with prescribed coefficients over a finite field

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field ${\mathbb{F}}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for all but $d^2-d-1$ values of $s$ in ${\mathbb{F}}_q$, the probability that $f(T)+sT+b$ is irreducible over ${\mathbb{F}}_q$ (as $b\in {\mathbb{F}}_q$ is chosen uniformly at random) is $1\mathrm{/}d+O(q^{-1\mathrm{/}2})$.