A polynomial analogue of Jacobsthal function

For a polynomial $f(x)\in \mathbb Z[x]$ we study an analogue of Jacobsthal function defined by $j_f(N) =\max_{m}\{$for some $x \in \mathbb N$ the inequality $(x+f(i),N)>1$ holds for all $i\leqslant m\}$. We prove the lower bound
$$ j_f(P(y))\gg y(\ln y)^{\ell_f-1}\biggl(\frac{(\ln\ln y)^2}{\ln\ln\ln y}\biggr)^{h_f}\biggl(\frac{\ln y\ln\ln\ln y}{(\ln\ln y)^2}\biggr)^{M(f)}, $$
where $P(y)$ is the product of all primes $p$ below $y$, $\ell_f$ is the number of distinct linear factors of $f(x)$, $h_f$ is the number of distinct non-linear irreducible factors and $M(f)$ is the average size of the maximal preimage of a point under a map $f\colon \mathbb F_p\to \mathbb F_p$. The quantity $M(f)$ is computed in terms of certain Galois groups.