I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory of Prym varieties, since the prym of a double cover of a q-t curve is the jacobian of an hyperelliptic curve B.

There are a few things that are not completely clear to me: first, can the two points p,q identified be one fiber of the hyperelliptic pencil on X?

second, can the hyperelliptic curve B be reconstructed (say, find its weierstrass points) starting from the data of the weierstrass points of X plus p and q?

Third, on the other hand the canonical model of a q-t curve sits on a rational normal cone, the singular point coinciding with the vertex of the cone. Do the ruling of the cone induce the hyperelliptic map on C? Is there a model of C on a Hirzebruch surface or one just needs to blow up the cone in the vertex?