 Affine principal-parts interpolation for a locally free sheaf

Let $X$ be an irreducible affine curve, $A=k[X]$, $K=\mathrm{Frac}A$, and $\mathcal{F}=\tilde{M}$ be locally free. Given finitely many classes in $M_K/M_{{\mathfrak{m}}_P}$, choose one nonzero $a\in A$ clearing denominators of representatives. The ring $A/(a)$ is Artinian and is the product of its finitely many local factors. Hence $M/aM\cong {⨁}_{P\in V(a)}(M/aM{)}_{{\mathfrak{m}}_P}$. Prescribe $a{v}_P$ modulo $a{M}_{{\mathfrak{m}}_P}$ at the requested points and zero at the other zeros of $a$. Lift this tuple to $w\in M$; the section $w/a$ has exactly the requested rational principal parts and no others. This proves surjectivity without assuming the curve is nonsingular.