  We study the dynamics of two-dimensional (2D) localized modes in the
nonlinear lattice described by the discrete nonlinear Schr\"{o}dinger (DNLS)
equation, including a local linear or nonlinear defect. Discrete solitons
pinned to the defects are investigated by means of the numerical continuation
from the anti-continuum limit and also using the variational approximation
(VA), which features a good agreement for strongly localized modes. The models
with the time-modulated strengths of the linear or nonlinear defect are
considered too. In that case, one can temporarily shift the critical norm,
below which localized 2D modes cannot exists, to a level above the norm of the
given soliton, which triggers the irreversible delocalization transition.
