Hi there, welcome to the forum.
I want to add a button in the panel which shows lyrics.
It's been a while since I had a look at plugins, but the last time I checked, the API does not provide access to any of MusicBee's native panels.
So, to answer your question: you cannot add your own button or make any change whatsoever to MusicBee's lyrics panel.
With that said, you can however attempt to recreate that lyrics panel as the API allows developers to create their own dockable panel/element.
Then from there, you would modify it however you want - it doesn't sound worth the trouble though.
What function would the button that you wanna add perform in that lyrics panel?