Can you copyright your own version of a cover song?

It sounds like you may be making a derivative work. That is a work that is closely based on the original, but changes it. The right to make a derivative work lies with the copyright holder. For songs, in most cases, that is the songwriter or publisher. To make a derivative work, you need to get permission from the copyright holder. A song becomes a derivative work if you change the lyrics of significantly change the music. You might be able to copyright your version of the song with yourself named as the arranger. Also, once you record the song, you can of course register copyright on your sound recording.

No, you cannot hold the copyright in a cover song because it is a derivative work of the original song writer's copyright. You can however seek to license the song if you wish to release your own version for profit, but it will require the copyright holder's consent and likely a fee paid.

First, your version as a cover is called a derivative work. There are two major kinds of copyright. There's a composition copyright in the performance copyright. While you may have the ability to copyright your individual performance of a cover song you would still need to obtain the appropriate licensing from the copyright holder of the original composition. However, because your cover is based on the original work of someone else, you would not be able to copyright your composition.

Music has two types of copyright: one on the composition and one on the performance. So as long as you've got a license (or some other form of permission) on the composition, you can copyright your own performance.