How should I approach this unfinished research that I do not want to do?
Talk to your supervisor when you know how long you will need to finish the publication.

You are currently in the phase of a project where you have to make very sure everything really works. This is usually the least motivating stage, but it is absolutely necessary. So your supervisor will (hopefully) push you to go through this phase and finish the publication, because finishing research projects is something you have to learn as a budding researcher.

Of course, not every project should be finished. If the costs (time invested) of a project outweigh the benefits (results and publication strength), your supervisor may want to abandon the project. It is important to find out how long you still need, and only then (if still applicable - maybe it's a two-week bug fix that doesn't feel like drinking bleach) have a discussion with your supervisor. You can bring up your independent research during that discussion as well - either as an alternative or as a next project.