  We study the existence and the properties of ground states at fixed mass for
a focusing nonlinear Schr\"odinger equation in dimension two with a point
interaction, an attractive Coulomb potential and a nonlinearity of power type.
We prove that for any negative value of the Coulomb charge, for any positive
value of the mass and for any L$^2$-subcritical power nonlinearity, such ground
states exist and exhibit a logarithmic singularity where the interaction is
placed. Moreover, up to multiplication by a phase factor, they are positive,
radially symmetric and decreasing. An analogous result is obtained also for
minimizers of the action restricted to the Nehari manifold, getting the
existence also in the L$^2$-critical and supercritical cases.
