Because you are only interested in a neighborhood of $R$, you might assume that $M$ is (connected and) noncompact. A function $f$ satisfies $\partial \bar{\partial} f=0$ iff it is harmonic. Now pick an open cover $(U_i)$ of $M$ and local solutions $\phi_i$ of your problem.
The differences $\phi_{ij}:=\phi_i - \phi_j$ are harmonic and form a cocycle, representing a cohomology class in $H^1 (M; \mathcal{H})$, where $\mathcal{H}$ is the sheaf of harmonic functions.

There is a short exact sequence of sheaves $0 \to \mathbb{R} \to \mathcal{O} \to \mathcal{H} \to 0$ (take real parts). Now $H^1 (M; \mathcal{O})=0$ (a deep result, holds for noncompact Riemann surfaces) and $H^2 (M;\mathbb{R})=0$ (not so hard, also since $M$ is noncompact. Thus $H^1 (M; \mathcal{H})=0$.

Thus there are harmonic functions $f_i$ on $U_i$ with $f_i - f_j = \phi_{ij}$. The functions $\phi_i - f_i$ still are local solutions for your problem, and they coincide on intersections, therefore define a global solution.