Thanks for reaching out about this here in the Community.
As @user-removed mentioned, Spotify should continue to play in the background when you're screen is locked without waking it up. We suggest to check on your phone's settings > notifications > Spotify and make sure to disable any option to keep your screen on while locked.
Let us know how it goes. We'll be here in case you have any questions.
Help others find this answer and click "Accept as Solution".
If you appreciate my answer, maybe give me a Like.