 Solution

Take pairwise disjoint coordinate discs $U_m$ around $x_m$ and put $U_0=S\setminus \{x_1,\dots ,x_N\}$. A prescribed principal part of a meromorphic function $P_m$ is holomorphic on $U_0\cap U_m$. Definewith all other components zero. No three distinct cover members meet, so the Čech cocycle condition is automatic. Thus $(P_{ij})$ is a Čech one-cocycle for $\mathcal{O}$.