It's okaaaay~ Relax~
-It will definitely not kill him as long as it's taken care of nicely. It WILL heal in time as long as his water is not too dirty or infectious.
- Do REALLY GOOD water changes. Maybe more often than you do, depends on the wound. Water changes always help
-If you want to, use aquarium salt. I use it, but if it's a small cut, i won't.
-You can also use medication, but I always go a natural route.
-i feed my bettas pellets soaked in garlic juice, and it's proven to help heal their fins. All bettas seem to adore garlic juice as i've yet to see one reject it.
Get rid of the aquarium plant if you think it's the culprit, and then replace it with a silk/live plant.