Apple Music skip like a scratched CD on my device

Fix:

1. Settings → Music
2. Audio Quality → Streaming
3. Change from Lossless / Hi-Res Lossless → High Quality (AAC)

If you are streaming, you may have a poor Internet connection. Changing to Hi-Res as was suggested in another reply would only make that worse, or no change at best.

Unfortunately you don't tell us if this is streamed music, or music actually on your device. If it is music on your device, do the original also skip? It could the the files on your device are corrupt.