If $f \colon C \to C'$ is a dominant morphism of smooth projective curves, there is a norm map $f_\ast = \mathrm{Nm} \colon JC \to JC'$ between their Jacobians, and we can consider the abelian subvariety $Z = (\ker \mathrm{Nm})^0$ and the polarization on $Z$ induced from $JC$. In two particularly interesting cases this polarization is twice a principal polarization $\Xi$, namely when $f$ has degree two and is either unramified or has two ramification points. In this case, one calls the ppav $(Z,\Xi)$ the Prym variety of $f$.

This construction works just as well in families, so it defines morphisms between some moduli spaces. If $R_g$ denotes the moduli space of smooth genus $g$ curves $C'$ and an unramified connected double cover $C \to C'$, one gets a Prym map $R_g \to A_{g-1}$. Here $A_g$ is the moduli space of dimension $g$ principally polarized abelian varieties. This map is rather well studied (e.g. the papers of Beauville, Donagi-Smith, Donagi and too many others to mention here).

On the other hand, one could also let $R_{g,2}$ be the moduli space of smooth genus $g$ curves with a double cover with two branch points. One gets in the same way a Prym map $R_{g,2} \to A_g$. But I have never seen any paper dealing with the properties of this map; maybe this is because of my incomplete knowledge of the literature. (One reason it may be less interesting is that unlike the ordinary Prym map in genus six, this one should never be generically finite for any $g$.)

Here are for instance some natural questions about this map: Is it dominant when $g \leq 4$? If yes, what is the structure of the generic fiber? Is it generically injective when $g \geq 5$? Does it extend to a compactification $\overline R_{g,2}$ using admissible covers, either by mapping to the Satake compactification or to a toroidal compactification like the 2nd Voronoi?

Is any of this known?