  The purpose of this article is to extend the uniqueness results for the two
dimensional Calder\'on problem to unbounded potentials on general geometric
settings. We prove that the Cauchy data sets for Schr\"odinger equations
uniquely determines potentials in $L^{p}$ for $p> 4/3$. In doing so, we first
recover singularities of the potential, from which point a $L^2$-based method
of stationary phase can be applied. Both steps are done via constructions of
complex geometric optic solutions and Carleman estimates.
