Tracks listed under "Manually Added Songs" continue to re-appear
Solution 1:
I had the exact same problem with my iphone and itunes account. I would remove the manually added songs but whenever I synced they magically re-appeared. It took me some time but I figured it out eventually. If you go to your itunes library and untick the unwanted albums/tracks and THEN remove the manually added songs and hit apply, all the unwanted songs are removed from the device.
Solution 2:
When you delete songs on an iPod (or other iOS device), they get copied back the next time you sync with iTunes. This makes no sense, of course, but that's the way it works. (It correctly does not re-sync apps you delete, most of the time.) The only way around it is to un-sync the tracks - make sure they are not selected in the Music tab, or are not in any playlists, or uncheck the tracks and choose the Only sync checked songs and videos options. Or you can manually sync music to the iOS device, but that may not be what you want.
Solution 3:
I took the longest time trying to figure this out, but finally came to a solution I hadn't seen in any of my research, so I thought I'd better share.
I'm on iOS 10 and haven't been able to toggle iCloud music on my iPhone, which I thought was where the problem was coming from, but it isn't. The solution is actually fairly simple:
First, on your iPhone, go to Settings>Music and select "downloaded music". Find the songs that keep showing up in the "manually added" section and delete them (or if you're like me, delete all the songs so you can start fresh).
Then, on the computer, remove all of the "manually added" songs. Sync again, and the songs stay off!
It's finally taken off the songs that I'd initially bought on the phone but only wanted parts of (like a movie soundtrack that I'd only wanted the songs from, and not the instrumentals).
I hope this is helpful to anyone else with the same problem.
(Note to any responses: I have no plans on returning to this page now that I've found a solution, so anything directed toward me will go unanswered).