A generalization of the notion of spanning tree congestion for weighted graphs is introduced. The $L^p$ congestion of a spanning tree is defined as the $L^p$ norm of the edge congestion of that tree. In this context, the classical congestion is the $L^\infty$-congestion. Explicit estimations of the minimal spanning tree $L^p$ congestion for some families of graphs are given. In addition, we introduce a polynomial-time algorithm for approximating the minimal $L^p$-congestion spanning tree in any weighted graph and another two similar algorithms for weighted planar graphs. The performance of these algorithms is tested in several graphs.