A large integer is a sum of two prime avoiding numbers

Let $f(n)={\min }_p\mathrm{\mid }n-p\mathrm{\mid }$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1+n_2$ and $f(n_i)\gg \ln N(\ln \ln N{)}^{\delta }$, $i=1,2$.