We obtain improved distributed algorithms in the CONGEST message-passing setting for problems on power graphs of an input graph $G$. This includes Coloring, Maximal Independent Set, and related problems. We develop a general deterministic technique that transforms R-round algorithms for $G$ with certain properties into $O(R \cdot Δ^{k/2 - 1})$-round algorithms for $G^k$. This improves the previously-known running time for such transformation, which was $O(R \cdot Δ^{k - 1})$. Consequently, for problems that can be solved by algorithms with the required properties and within polylogarithmic number of rounds, we obtain {quadratic} improvement for $G^k$ and {exponential} improvement for $G^2$. We also obtain significant improvements for problems with larger number of rounds in $G$.