  It is well known that the kernel of a Toeplitz operator is nearly invariant
under the backward shift $S^*$. This paper shows that kernels of finite-rank
perturbations of Toeplitz operators are nearly $S^*$-invariant with finite
defect. This enables us to apply a recent theorem by
Chalendar--Gallardo--Partington to represent the kernel in terms of backward
shift-invariant subspaces, which we identify in several important cases.
