Such a 2 point ramified double cover occurs from an unramified one when a loop shrinks to a point, which... let's see now, does intersect [I believe] the loop on which the double cover is based. Then I believe the induced double cover of the normalized curve is ramified at the 2 points over the singular point.

Vice versa, 2 point ramified double covers are a subcase of Beauville's "star - double covers" of curves with one node. E.g. 2 point ramified covers of genus 5 curves are a special case of Beauville's "star - double covers" of genus 6 curves with one node.

Anyway such ramified double covers have thus been studied by Wirtinger and also explicitly by Fay, I believe, as well as more generally by Beauville.

This is not as up to the minute as the beautifully detailed thesis linked in the answer above, but just a couple of historical comments.

Such a 2 point ramified double cover occurs from an unramified one when a loop shrinks to a point, which... let's see now, does intersect [I believe] the loop on which the double cover is based. Then I believe the induced double cover of the normalized curve is ramified at the 2 points over the singular point.

Vice versa, 2 point ramified double covers are a subcase of Beauville's "star - double covers" of curves with one node. E.g. 2 point ramified covers of genus 5 curves are a special case of Beauville's "star - double covers" of genus 6 curves with one node.

Anyway such ramified double covers have thus been studied by Wirtinger in the 19th century, and also explicitly by Fay in his book on theta functions, as well as more generally by Beauville.

Such a 2 point ramified double cover occurs from an unramified one when a loop shrinks to a point, which... let's see now, does intersect [I believe] the loop on which the double cover is based. Then I believe the induced double cover of the normalized curve is ramified at the 2 points over the singular point.

Vice versa, 2 point ramified double covers are a subcase of Beauville's "() - double covers" of curves with one node. E.g. 2 point ramified covers of genus 5 curves are a special case of Beauville's () - double covers of genus 6 curves with one node.

Anyway such ramified double covers have thus been studied by Wirtinger and also explicitly by Fay, I believe, as well as more generally by Beauville.

Such a 2 point ramified double cover occurs from an unramified one when a loop shrinks to a point, which... let's see now, does intersect [I believe] the loop on which the double cover is based. Then I believe the induced double cover of the normalized curve is ramified at the 2 points over the singular point.

Vice versa, 2 point ramified double covers are a subcase of Beauville's "star - double covers" of curves with one node. E.g. 2 point ramified covers of genus 5 curves are a special case of Beauville's "star - double covers" of genus 6 curves with one node.

Anyway such ramified double covers have thus been studied by Wirtinger and also explicitly by Fay, I believe, as well as more generally by Beauville.