Hi
Yes, it can.
Most - but not all - Bluetooth headphones that use the A2DP Bluetooth profile can be used with Apple Watch for listening to music from it:
- Listen to music on your Apple Watch - Apple Support
- Use Bluetooth accessories with Apple Watch - Apple Support
Apple does not publish a list of compatible headphones.
Headphones that are least likely to be compatible are those that also offer additional built-in functions such as fitness tracking, a heart rate monitor and/or a built-in media player.
For any such headphones, I suggest that you check with the manufacturer before making a purchase.
You may wish to consider buying AirPods, Beats X or other Apple / Beats products that are able to connect to your iPhone and Apple Watch at the same time, switching between them automatically. For example, in the US:
Headphones & Speakers - Watch Accessories - Apple
In respect of playback issues, it may help to check for any potential sources of wireless interference:
Potential sources of Wi-Fi and Bluetooth interference - Apple Support