  Let $q$ be a prime. Let $G$ be a residually finite group satisfying an
identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$
such that the element $x^m$ is a bounded Engel element. We prove that $G$ is
locally virtually nilpotent. Further, let $d,n$ be positive integers and $w$ a
non-commutator word. Assume that $G$ is a $d$-generator residually finite group
in which all $w$-values are $n$-Engel. We show that the verbal subgroup $w(G)$
has $\{d,n,w\}$-bounded nilpotency class.
