Weak Kerr-type nonlinearity is known to destroy Anderson localization in Hermitian disordered lattices, leading to the spreading of initially localized wave packets. However, the interplay of disorder, nonlinearity, and asymmetric transport in non-Hermitian systems remains largely unexplored. Here, we investigate this question using a nonlinear Hatano–Nelson model with onsite disorder. Starting from an Anderson-localized regime in the linear limit, we examine the time evolution of an initially localized excitation. While the linear system exhibits saturation of the second moment, confirming localization, the introduction of nonlinearity leads to unbounded growth, with a subdiffusive regime followed by super-ballistic spreading, signaling delocalization. Alongside the second moment, higher-order moments reveal that nonlinear spreading in the non-Hermitian system is accompanied by a significant reshaping of the wave packet, reflecting the interplay between asymmetric transport and interaction-induced mode coupling. Our results demonstrate that nonlinearity not only destroys Anderson localization but also qualitatively modifies the structure of the evolving wave packet in non-Hermitian disordered lattices.