Here's the process I follow for buffing out scratches. You should /definitely test it on an inconspicuous spot first/. It works best for harder plastics. Soft plastics like polycarbonate are considerably more difficult.
1: put tape around the scratched area, to limit the area affected by step 2, 2: sanding the area. Start with a low-ish grit like 320 or 400 if it's deep, but if it's a light scratch you can try higher grit like 1000 or 1500. 3: move the tape a bit farther out, and sand again at a higher grit. Repeat till you get to like 2500 grit 4: move the tape out one last time, then buff the plastic with a polishing compound like Novus 2 and a microfiber or lint free wipe all. Don't use paper towels. 5: remove the tape and continue buffing lightly to feather out the edges to make your polished spot less conspicuous. It's always going to flash a little bit from some angles and there's going to be a bit of visual distortion since it has less material in that spot now but ideally it should be mostly invisible.
For a real light scuff just skip straight to Novus 2 or similar polishing compounds. It's also good for polishing car headlight lenses that have gotten cloudy. Absolutely never for eyeglass lenses though. It'll destroy those.