Q: I can't delete a duplicate purchased song on my iphone 5c

Does this song appear to have a cloud symbol and arrow beside it?

Then the song is not on your device, it is available in the cloud. You can hide music that is not on your device by going to Settings>Music>Show All Music and turn that off.

So are you trying to get rid of both songs? The cloud is by one since it not on the device. I'm not sure you can delete that duplicate, but I'm checking.

I found this, Find and remove duplicate items in your iTunes library - Apple Support  but it refers to your iTunes library on a Mac. I'm guessing you can address it in your iTunes library on whatever computer OS you may be using. See if that helps you.