Throughout, I want my ambient space $X$ to be a smooth projective variety of dimension at least two.
I believe we can at least say the following. If $\mathcal{E}$ is a pure one-dimensional coherent sheaf on $X$, and the support of $\mathcal{E}$ is a smooth projective curve $C \subset X$, then $\mathcal{E}$ is the pushforward of a vector bundle on $C$. I believe this is true because pure sheaves are torsion-free restricted to their support, and torsion-free sheaves on a smooth projective curve are locally-free.
Now, removing the assumption that the support be smooth, I am wondering to what extent (if any) pure one-dimensional coherent sheaves on smooth projective varieties are pushforwards of vector bundles. Given a pure one-dimensional sheaf $\mathcal{E}$ supported on $C$, if we compose the normalization with the inclusion
$$\widetilde{C} \rightarrow C \hookrightarrow X,$$
can we always realize $\mathcal{E}$ as the pushforward of a locally-free sheaf on $\widetilde{C}$ to $X$? What if $\mathcal{E}$ has scheme-theoretic support on a non-reduced curve $C$?
