Summary: Let \(K\) be an algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value, and let \(\phi \in K(z)\) with \(\deg (\phi )\ge 2\). Recently, Rumely introduced a family of discrete probability measures \(\{\nu _{\phi ^n}\}\) on the Berkovich line \(\mathbf{P }^1_{\text{K}}\) over \(K\) which carry information about the reduction of conjugates of \(\phi \). In a previous article, the author showed that the measures \(\nu _{\phi ^n}\) converge weakly to the canonical measure \(\mu _\phi \). In this article, we extend this result to allow test functions which may have logarithmic singularities at the boundary of \(\mathbf{P }^1_{\text{K}}\). These integrands play a key role in potential theory, and we apply our main results to show the potential functions attached to \(\nu _{\phi ^n}\) converge to the potential function attached to \(\mu _\phi \), as well as an approximation result for the Lyapunov exponent of \(\phi \).