A polynomial analogue of Jacobsthal function

For a polynomial $f(x)\in \mathbb{Z}[x]$ we study an analogue of Jacobsthal function, defined by the formula $$j_f(N)=\underset{m}{\max }\{\text{For some }x\in \mathbb{N}\text{ the inequality }(x+f(i),N)>1\text{ holds for all }i\le m\}\mathrm{.}$$ We prove a lower bound $$j_f(P(y))\gg y(\ln y{)}^{{\mathrm{\ell }}_f-1}{\left(\frac{(\ln \ln y{)}^2}{\ln \ln \ln y}\right)}^{h_f}{\left(\frac{\ln y\ln \ln \ln y}{(\ln \ln y{)}^2}\right)}^{M(f)},$$ where $P(y)$ is the product of all primes $p$ below $y$, ${\mathrm{\ell }}_f$ 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 $f:{\mathbb{F}}_p\to {\mathbb{F}}_p$. The quantity $M(f)$ is computed in terms of certain Galois groups.