When I ask this, I don't mean a personal song, but for example: I want to listen to Adele's "When We Were Young" (Live at The Church Studios) specifically, but it's not on the Spotify database. Is there any way it can be uploaded so I can get it on my playlist?
There is no such thing like a cloud service where you can upload your songs for future listening. Please refer to the options menu where you can add local data to the catalogue. I assume that you have the particular song saved on your device(s) since you asked for an upload option?