The distance to an irreducible polynomial, II

P. Turán asked if there exists an absolute constant $C$ such that for every polynomial $f\in \mathbb{Z}[x]$ there exists an irreducible polynomial $g\in \mathbb{Z}[x]$ with $\deg (g)\leq \deg (f)$ and $L(f-g)\leq C$, where $L(\cdot )$ denotes the sum of the absolute values of the coefficients. We show that $C=5$ suffices for all integer polynomials of degree at most $40$ by investigating analogous questions in $\mathbb{F}_p[x]$ for small primes $p$. We also prove that a positive proportion of the polynomials in $\mathbb{F}_2[x]$ have distance at least $4$ to an arbitrary irreducible polynomial.