= Solution
Let $\mathcal P=\mathcal M_X/\mathcal O_X$, the <sheaf of meromorphic principal parts>. Exactness of taking <stalks> gives
$$
\mathcal P_x=\mathcal M_{X,x}/\mathcal O_{X,x}.
$$
Choose a <holomorphic coordinate> $t$ vanishing at $x$. Every meromorphic germ has a finite negative tail in its <Laurent series>; the holomorphic tail vanishes in the quotient. Hence
$$
\boxed{\mathcal P_x\cong\bigoplus_{m\geq1}\mathbb C\,t^{-m}.}
$$
This coordinate description is a vector-space identification; the intrinsic space is the quotient of germs, so a coordinate change need not preserve the displayed basis.
For each point inclusion $i_x:\{x\}\to X$, a principal part defines a local meromorphic germ near $x$, whose quotient class is zero off $x$. Gluing with zero on the complement gives a map from its <skyscraper sheaf> to $\mathcal P$. These maps induce an isomorphism on every stalk and therefore an isomorphism of sheaves:
$$
\boxed{\mathcal P\cong\bigoplus_{x\in X}(i_x)_*\mathcal P_x.}
$$
The <direct sum of sheaves> means the sheafification of the sectionwise direct-sum presheaf. Its sections can have infinitely many nonzero components globally, but their point supports form a <locally finite family of subsets>. This matches the fact that poles of a <meromorphic function> are locally finite. On a compact <Riemann surface> only finitely many points can occur; on a noncompact one a discrete infinite family is allowed.
For a global principal-part family $P$, choose local meromorphic lifts $m_i$ on a sufficiently small <open cover>. The differences $m_j-m_i$ are <holomorphic functions> and form a <Čech cocycle>. The <connecting homomorphism> sends $P$ to the resulting class in $H^1(X,\mathcal O_X)$. If this class vanishes, after refining the cover write $m_j-m_i=h_j-h_i$; then the functions $m_i-h_i$ glue to a global <meromorphic function>. Conversely any global lift makes the class zero. Equivalently, exactness of the given <long exact sequence in sheaf cohomology> says
$$
\boxed{\delta(P)=0\iff\text{there is a global meromorphic function with exactly the prescribed principal parts}.}
$$
\b[The connecting class is the obstruction to the Mittag-Leffler problem on a Riemann surface.] It rules on the specified poles and their finite negative Laurent tails, with no additional poles allowed. When a solution exists, any two solutions differ by a global <holomorphic function>.
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