The vertex-edge Wiener index of a simple connected graph GG is defined as the sum of distances between vertices and edges of GG. The vertex-edge Wiener polynomial of GG is a generating function whose first derivative is a q−q−analog of the vertex-edge Wiener index. Two possible distances D1(u,e|G)D1(u,e|G) and D2(u,e|G)D2(u,e|G) between a vertex uu and an edge ee of GG can be considered and corresponding to them, the first and second vertex-edge Wiener indices of GG, and the first and second vertex-edge Wiener polynomials of GG are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.