<p>Two-dimensional non-Hermitian photonic lattices with asymmetric couplings offer rich possibilities for controlling wave localization, through the emergence of the non-Hermitian skin effect at lattice corners or sides. Yet, how optical nonlinearity modifies these boundary-localization characteristics remains largely unexplored. Here we show that in a two-dimensional Hatano-Nelson lattice with Kerr nonlinearity, the interplay between self-trapping and directional propagation leads to position-dependent amplitude thresholds. Single-site excitations having above a critical amplitude become confined to their initial position, with lower thresholds near the position where the linear eigenmodes are localized and higher thresholds within the lattice’s bulk. Additionally, we study the differences of this dynamical interplay, for wider initial excitations, between the focusing and defocusing Kerr-nonlinearity regimes. Lastly, we identify skin soliton solutions in a variety of two-dimensional lattice geometries featuring coupling asymmetry. This work paves the way for future investigations regarding transport and soliton formation in higher-dimensional nonlinear non-Hermitian lattices.</p>