  In this paper, we study a natural discretization of the smooth Gaussian
curvature on surfaces. A discrete uniformization theorem is established for
this discrete Gaussian curvature. We further investigate the prescribing
combinatorial curvature problem for a parametrization of this discrete Gaussian
curvature, which is called the combinatorial $\alpha$-curvature. To find
decorated piecewise hyperbolic metrics with prescribed combinatorial
$\alpha$-curvatures, we introduce the combinatorial $\alpha$-Ricci flow for
decorated piecewise hyperbolic metrics. To handle the potential singularities
along the combinatorial $\alpha$-Ricci flow, we do surgery along the flow by
edge flipping under the weighted Delaunay condition. Then we prove the longtime
existence and convergence of the combinatorial $\alpha$-Ricci flow with
surgery. As an application of the combinatorial $\alpha$-Ricci flow with
surgery, we give the existence of decorated piecewise hyperbolic metrics with
prescribed combinatorial $\alpha$-curvatures. We further introduce the
combinatorial $\alpha$-Calabi flow with surgery and study its longtime
behavior.
