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The answer is positive: there is a surjective, generically finite morphism $\text{sym}^d(C)\to \text{Prym}(C/C')$, at least away from small characteristics. Fix a $k$-point $x$ of $C$, and use that to define an Abel map, $\alpha_x:C \to J(X)$. The induced composition morphism $$C^g \xrightarrow{\alpha^g} J(C)^g \xrightarrow{\Sigma} J(C),$$ is surjective ...