  We provide $q$-moment estimates on annuli for weak solutions of the singular
$p$-Laplace equation where $p$ and $q$ are conjugates. We derive $q$-uniform
integrability for some critical parameter range. As a application, we derive a
mass conservation as well as a weak convergence result for a larger critical
parameter range. Concerning the latter point, we further provide a rate of
convergence of order $t^{q-1}$ of the solution in the $q$-Wasserstein distance.
