  In the classical Hardy space $H^2(\mathbb{D})$, it is well-known that the
kernel of the Hankel operator is invariant under the action of shift operator S
and sometimes nearly invariant under the action of backward shift operator
$S^{*}$. It appears in this paper that kernels of finite rank perturbations of
Hankel operators are almost shift invariant as well as nearly $S^*$- invariant
with finite defect. This allows us to obtain a structure of the kernel in
several important cases by applying a recent theorem due to Chalendar,
Gallardo, and Partington.
