  A left Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$
such that for every $x\in G$ all sufficiently long commutators
$[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is a left
Engel element precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) We prove
that if a finite group $G$ admits an automorphism $\varphi $ of prime order
coprime to $|G|$ such that for some positive integer $m$ every element of the
centralizer $C_G(\varphi )$ has a left Engel sink of cardinality at most $m$,
then the index of the second Fitting subgroup $F_2(G)$ is bounded in terms of
$m$.
  A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr
R}(g)$ such that for every $x\in G$ all sufficiently long commutators
$[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right
Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) We prove
that if a finite group $G$ admits an automorphism $\varphi$ of prime order
coprime to $|G|$ such that for some positive integer $m$ every element of the
centralizer $C_G(\varphi )$ has a right Engel sink of cardinality at most $m$,
then the index of the Fitting subgroup $F_1(G)$ is bounded in terms of $m$.
