Approximability of word maps by homomorphisms

Generalizing a recent result of Mann, we show that there is an explicit function $f:(0,1]\to (0,1]$ such that for every reduced word $w$, say in $d$ variables, there is an explicit reduced word $v$ in at most $3d$ variables (nontrivial if the length of $w$ is at least $2$) such that for all $\rho \in (0,1]$, the following holds: If $G$ is any finite group for which the word map $w_G:G^d\to G$ agrees with some fixed homomorphism $G^d\to G$ on at least $\rho \mathrm{\mid }G{\mathrm{\mid }}^d$ many arguments, then the word map $v_G:G^{3d}\to G$ has a fiber of size at least $f(\rho )\mathrm{\mid }G{\mathrm{\mid }}^{3d}$. We also discuss some applications of this result.