  Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume
that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union
contains all $w$-values in $G$. Assume further that all elements of the
subgroups $G_1,\ldots,G_m$ are $n$-Engel in $G$. It is shown that the verbal
subgroup $w(G)$ is $s$-Engel for some $\{m,n,w\}$-bounded number $s$.
