Differing opinions on water changes but I'd personally always do extra water changes if I saw an injury/illness. A 50% change every day shouldn't create any instabilitites or fish issues IMO unless the tank is improperly set up, or you have drastically different stats in your tap water compared to tank water, and also have very poor quaility tap water for fishkeeping...at which point I'd be looking into RO or other filters. Obviously this may not apply to some very sensitive wild fish with unusual water stat requirements, but domestic bettas should thrive with very large frequent water changes (they are commonly stored in cups/bags that get 100% of the water changed every day, which is a terrible practice, but shows thier resiliance). So don't be scared of big water changes
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Melafix is quite good for external wounds, but be careful with bettas as they are labyrinth fish that breath air, and Melfix has been known to cause issues due to the tea tree oil vapour that sits in a layer above the water surface and can hurt the betta when they go up for a breath. It also makes the water surface foamy which makes it harder for bettas to take in fresh air. They actually make a version called "Bettafix" to prevent this, but I don't know how it differs chemically, and IME Indian Almond leaves are much more useful and effective for keepings bettas healthy anyway.
Oak leaves will be better than nothing if you can find some clean ones locally (ideally not from the side of a busy road), Indian Almond Leaves from eBay or a LFS would be more effective and safer than Melafix if you can can get them though. Alder cones could be useful too if you happen to be able to collect some locally. Also I wouldnt advise actually boiling the leaves, doing so would remove most of the beneficial tannins you are adding them to provide in the first place. A dry leaf is not going to have any fish pathogens on it to worry about, if you are really worried you could quickly rinse them in a sieve with boiling water from the kettle, but don't steep them.
I wouldn't worry too much though! From the pictures, it looks like he has just lost one of the scales from his gill cover (operculum). I would guess he got his face stuck in somewhere searching for food and just scraped it - should heal quickly as long as the water is clean
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