Retrieved iTunes Store purchases not syncing to Apple Music

So to make a long story short, I accidentally deleted all of my music files from my computer, recycle bin and all, but luckily, I was able to retrieve most of them back from my iPod.

The problem being the stuff that I purchased on the iTunes Store is still marked down as purchased as in when I look it up on the store it doesn’t give me the option to buy it, but it won’t sync to my Apple Music, even with my account authorized on the computer, and even when moving the purchased files onto the purchased playlist that iTunes automatically makes for you. Is there a way to fix this? If so how? I don’t want my money to go to waste and I want to have the option to listen to my purchased music on Apple Music too.

[Re-Titled by Moderator]

Original Title: Itunes Files