The nonlinear Hall effect has recently attracted significant interest due to
its potentials as a promising spectral tool and device applications. A theory
of the nonlinear Hall effect on a disordered lattice is a crucial step towards
explorations in realistic devices, but has not been addressed. We study the
nonlinear Hall response on a lattice, which allows us to introduce disorder
numerically and reveal a mechanism that was not discovered in the previous
momentum-space theories. In the mechanism, disorder induces an increasing
fluctuation of the nonlinear Hall conductance as the Fermi energy moves from
the band edges to higher energies. This fluctuation is a surprise, because it
is opposite to the disorder-free distribution of the Berry curvature. More
importantly, the fluctuation may explain those unexpected observations in the
recent experiments. We also discover an "Anderson localization" of the
nonlinear Hall effect. This work shows an emergent territory of the nonlinear
Hall effect yet to be explored