  In hyperbolic background geometry, we investigate a generalized circle
packing (including circles, horocycles and hypercycles) with conical
singularities on a surface with boundary, which has a total geodesic curvature
on each generalized circle of this circle packing and a discrete Gaussian
curvature on the center of each dual circle. The purpose of this paper is to
find this type of circle packings with prescribed total geodesic curvatures on
generalized circles and discrete Gaussian curvatures on centers of dual
circles. To achieve this goal, we firstly establish existence and rigidity on
this type of circle packings by the variational principle. Secondly, for $p>1$,
we introduce combinatorial $p$-th Calabi flows to find the circle packing with
prescribed total geodesic curvatures on generalized circles and discrete
Gaussian curvatures on centers of dual circles for the first time.
