  We formulate the Bergman-type interpolation problem on finite open Riemann
surfaces covered by the unit disk. Our version of the interpolation problem
generalizes Bergman-type interpolation problems previously studied by Seip,
Berntsson, Ortega Cerd\`a, and a number of other authors. We then prove
necessary and sufficient conditions for interpolation, and also some sufficient
conditions under even weaker hypotheses. The results extend work of Ortega
Cerd\`a, who resolved the case in which the boundary of the surface is pure
$1$-dimensional. Our version of the interpolation problem effectively changes
the geometry of the underlying space near the punctures, thereby linking in a
crucial way with the previous article in this two-part series.
