DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2017. In this paper, we prove that every planar graph without 4-cycles adjacent to k-cycles is DP-4-colorable for k=5 and 6. As a consequence, we obtain two new classes of 4-choosable planar graphs. We use identification of vertices in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.