  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 profinite group $G$ admits a coprime automorphism $\varphi $ of prime
order such that every fixed point of $\varphi $ has a finite right Engel sink,
then $G$ has an open locally nilpotent subgroup.
  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 profinite group $G$ admits a coprime automorphism $\varphi $ of prime
order such that every fixed point of $\varphi $ has a finite left Engel sink,
then $G$ has an open pronilpotent-by-nilpotent subgroup.
