How exactly is it that you are trying to play these songs? First, make sure you are on the latest firmware on both your Apple Watch and iPhone. I have had issues with Apple Music simply because I haven't updated. If you are on the latest version of iOS and watchOS, you could try re-pairing your watch to your iPhone.
First, you'll have to unpair your Apple Watch from the Watch itself:
Unpair your Apple Watch and iPhone - Apple Support
Doing this with your iPhone in range will also backup your watch, so you needn't worry about losing information.
Then, go ahead and re-pair it:
Set up your Apple Watch - Apple Support
A fresh start should see an end to this issue of yours, but if it still persists, I would contact Apple Support directly and ask them.