Let G be a graph whose edges are colored with k colors, and H=(H1,⋯,Hk) be a k-tuple of graphs. A monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic copy of Hi in color i, for some 1≤i≤k. Let φk(n,H) be the smallest number ϕ, such that, for every order-n graph and every k-edge-coloring, there is a monochromatic H-decomposition with at most ϕ elements. Extending the previous results of Liu and Sousa [Monochromatic Kr-decompositions of graphs, J Graph Theory 76 (2014), 89-100], we solve this problem when each graph in H is a clique and n≥n0(H) is sufficiently large.