Big water change (75%) and gravel clean the substrate every day for a week.
Monitor the damaged areas and if it goes red and inflamed it has a bacterial infection. If it goes white and fluffy it's a fungal infection. You can treat both with salt.
Wrap a sponge around the intake on the filter to stop him being sucked up.
The best filter for a Betta is an air operated sponge filter.
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SALT
You can add rock salt (often sold as aquarium salt), swimming pool salt, or any non iodised salt (sodium chloride) to the aquarium at the dose rate of 1 heaped tablespoon per 20 litres (5 gallons) of water. If there is no improvement after 48 hours you can double that dose rate so there is 2 heaped tablespoons of salt per 20 litres.
Keep the salt level like this for 1 week. If there's no improvement after a few days with salt stop using it and look for a broad spectrum medication that treats bacteria and fungus.
The salt will not affect the beneficial filter bacteria, fish, plants, shrimp or snails.
After you use salt and the fish have recovered, you do a 10% water change each day for a week using only fresh water that has been dechlorinated. Then do a 20% water change each day for a week. Then you can do bigger water changes after that. This dilutes the salt out of the tank slowly so it doesn't harm the fish.
If you do water changes while using salt, you need to treat the new water with salt before adding it to the tank. This will keep the salt level stable in the tank and minimise stress on the fish.
When you first add salt, add the salt to a small bucket of tank water (2 litres or 1/2 gallon) and dissolve the salt. Then slowly pour the salt water into the tank near the filter outlet. Add the salt over a couple of minutes.