Images of word maps with constants on algebraic groups

We study word maps with constants on quasisimple algebraic groups over a local field $L$. We prove that, for such a group $G=\mathbf{G}(L)$ and a word $w\in (G\ast\mathbf{F}_r)\setminus G$, either $w$ has a so-called Tomanov-small critical constant or the minimal dimension of the word image $w(G^r)\subseteq G$ of such a word is bounded from below by a function $c(G)>1$. We compute the optimal value of $c(G)$ for most quasisimple linear algebraic groups.