By introducing the notion of distributive constant for a family of closed subschemes, we establish a general form of the second main theorem for algebraic non-degenerate meromorphic mappings from a generalized $p$-Parabolic manifold into a projective variety with arbitrary families of closed subschemes. As its consequence, we give a second main theorem for such meromorphic mappings intersecting arbitrary hypersurfaces with an explicitly truncation level for the counting functions.