Considering the reciprocal function, it is sufficient to construct a holomorphic function with prescribed zeros, and prescribed finite portions of Taylor series at those zeros. For the plane and the unit disk this is an interpolation problem whose positive solution follows from Weierstrass's theorem (and its generalization to the unit disk). For arbitrary open Riemann surfaces, one can pass to the universal cover. So the answer to your question is positive on any open Riemann surface.

Considering the reciprocal function, it is sufficient to construct a holomorphic function with prescribed zeros, and prescribed finite portions of Taylor series at those zeros. For the plane and the unit disk this is an interpolation problem whose positive solution follows from Weierstrass's theorem (and its generalization to the unit disk).