Long strings of composite values of polynomials and a basis of order 2

We show that for any polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_1=\{n_1-m,\dots ,n_1+m\}$ and $I_2=\{n_2-m,\dots ,n_2+m\}$, such that $m=[(\log N)(\log \log N{)}^{1\mathrm{/}325525}]$, $I_1\cup I_2\subset [1,N]$, $N=n_1+n_2$, and $f(n)$ is composite for any $n\in I_1\cup I_2$. This extends the result in [5] which showed the same result but with $f(n)=n$.