  We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a
weak, static, uniform magnetic field perpendicular to the atomic plane. Within
the framework of the Rayleigh-Schr\"odinger perturbation theory, using the
Sturmian expansion of the generalized radial Coulomb Green function, we derive
explicit analytical expressions for corrections to an arbitrary planar
hydrogenic bound-state energy level, up to the fourth order in the strength of
the perturbing magnetic field. In the case of the ground state, we correct an
expression for the fourth-order correction to energy available in the
literature.
