  We show that for any polynomial $f$ from the integers to the integers, with
positive leading coefficient and irreducible over the rationals, if $x$ is
large enough then there is a string of $(\log x)(\log\log x)^{1/835}$
consecutive integers $n \in [1,x]$ for which $f(n)$ is composite. This improves
a result of the first author, Konyagin, Maynard, Pomerance and Tao, which
states that there are such strings of length $(\log x)(\log\log x)^{c_f}$,
where $c_f$ depends on $f$ and $c_f$ is exponentially small in the degree of
$f$ for some polynomials.
