How to sync deletion of imported song in Apple Music across devices?

I import songs to Apple Music on my desktop and let it sync to cloud so I have access to the songs across all my devices. Syncing imported songs works pretty much flawlessly, but deletion not so much. If I delete an imported song from my library using the desktop, every syncs immediately; but if I use my IPhone, the song is deleted for all my other devices (my IPad) except the desktop that I used to import the song. So if I listen to Apple Music on my desktop, there are a bunch of songs that I've long deleted still left in the library. If I check the file property of those songs, their cloud status says removed. Is there a way for me to quickly get rid of those already deleted songs from my desktop Apple Music library and make sure the future deleted songs will be removed from all libraries?