After the water change, give the filter a clean. Nothing drastic, and don't run anything under the tap, just take a bucket of the dirty tank water you removed, and give the filter media - sponges and any biological media etc, a quick rinse in the mucky water to remove the worst of the gunk. Don't try to get them spotless or anything, we want to keep your nitrifying bacteria alive, just squeeze the sponges in the water enough to remove most of the yuckiness, and gently swish the bio media around. Same if your filter uses cartridges, just swish them around and try to remove as much of the larger muck as you can.
If the cory is still eating, then he's still fighting!
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I know it's horrible to see them like that, believe me I know. There is a chance he won't make it I'm afraid, and I'm sorry about that. There are also times where euthanasia is the kinder option. But I don't think he's at that point. He hasn't had time yet to respond to treatment, and if the salt kills enough of that fungus, and he's in clean fresh water with zero ammonia/nitrites, and low nitrates, his immune system will have a chance to rally and help him fight off the infections, with the help of the salt reducing the amount of bacteria and fungus in the water and on his body. Fins can and do re-grow, and there are no nerves in his fins, so as horrible as it looks, the fin part at least isn't causing him pain... he's a poorly fish, but with the care you're giving him, he has a fighting chance.
The guppy I showed you yesterday looked terrible for the first 48 hours or so, I really thought I'd lose her. She was caught up in the sponge filter somehow when I lifted it out to clean, and she fell out and landed on the stand. Since I was right there and saw it happen I got her back in the tank right away, but somehow nearly all of her tail had been torn off right down to the body as you can see in the pics, and she also had damage all along one side, must have been the side that landed on the tank stand.
She had a hard time swimming normally at first, and by the next morning, the tail stump and that whole side were fungused, and she looked awful, really struggling to swim and looked stressed and exhausted, fin clamped etc. I used salt treatment and water changes, but I admit I didn't think she was going to recover. But she was still eating and still trying, and within 48 hours of salt treatment, the fungus had gone, and she wasn't fin-clamped anymore. Within a week, I could see her tail was beginning to grow back, and she made a full recovery. She was a sub-adult when it happened, and she ended up being one of my largest, healthiest females, with nothing but a couple of scars on her tail to show for it. So keep up the treatment, hope for the best while being prepared for the worst.
How's the guppy doing?