ITunes on my work computer
If you've subscribed to Apple Music, or if all the music you own is purchased, then all the music you have is the same on all of your Apple devices and yes, when you delete them from your Apple Music library then you'll lose the download of that song but it will always be available to you for as long as your subscription lasts.

If you've ripped CDs (or obtained content by other means) you'll need to copy those files from your work computer to external media and put them on your home computer. Usually iTunes stores its files in the Music folder on Windows and macOS (OS X), you can check where music is stored by going into Preferences in iTunes on the Advanced tab. Just place the music you copied from your work computer to your home one , then restart iTunes.