We investigate soliton mobility in the disordered Ablowitz-Ladik (AL) model and the standard nonlinear Schrödinger (NLS) lattice with the help of an effective potential generalizing the Peierls-Nabarro potential. This potential results from deviation from integrability, resulting of randomness for the AL model, and of both randomness and lattice discreteness for the NLS lattice. Statistical properties of such a potential are analyzed, and it is shown how the soliton mobility is affected by its size. The usefulness of this effective potential in studying soliton dynamics is demonstrated numerically. Further we propose two ways the soliton transport in presence of disorder can be enhanced: one is to use specific realizations of randomness, and the other one is to consider a specific soliton pair. ; Comment: 5 pages, 5 figures