  Recently, Liang and Partington \cite{YP} show that kernels of finite-rank
perturbations of Toeplitz operators are nearly invariant with finite defect
under the backward shift operator acting on the scalar-valued Hardy space. In
this article we provide a vectorial generalization of a result of Liang and
Partington. As an immediate application we identify the kernel of perturbed
Toeplitz operator in terms of backward shift-invariant subspaces in various
important cases by applying the recent theorem (\cite{CDP, OR}) in connection
with nearly invariant subspaces of finite defect for the backward shift
operator acting on the vector-valued Hardy space.
