  The nonlinear Thouless pumping is an exciting frontier of topological
physics. While recent works have revealed the quantized motion of solitons in
Thouless pumps, the interplay between the topology, nonlinearity and disorder
remains largely unexplored. Here, we investigate the nonlinear Thouless pumping
of solitons in the presence of an impurity in the context of a Bose-Einstein
condensate. Using both the Gross-Pitaevskii equation and Lagrangian variational
approach, we analyze the interaction between a moving soliton and an impurity.
Without the pump, the soliton can pass through a light impurity, but gets
trapped by the impurity with large mass. In marked contrast, we find the
soliton in Thouless pumps can always transit through the impurity, and its
motion is topologically quantized. Our result explicitly showcases the
robustness of topological soliton pumping against microscopic imperfections,
and opens a new perspective in the information processing with solitons.
