 Solution

If ${\check{H}}^1(S,\mathcal{O})=0$, that cocycle is a Čech coboundary: there are $f_i\in \mathcal{O}(U_i)$ withSet $F=-f_0$ on $U_0$ and $F=P_m-f_m$ on $U_m$. These formulas agree on overlaps, so the sheaf gluing axiom gives a global meromorphic function. Moreover $F-P_m=-f_m$ is holomorphic near $x_m$. Thus $F$ solves the Mittag-Leffler problem on a Riemann surface with precisely the prescribed principal parts.