Pondotaku, welcome to the forum.
Just some bullet points.
Cut any trees on the dam that are smaller than 4" DBH (diameter breast height). IMMEDIATELY paint the stump end with Tordon RTU (I use a chip brush in a coffee can that has Tordon RTU in it, it uses less than squirting it from the bottle).
As for the muck, if you can get an excavator in there to remove the muck from the area that you want it gone, I would do that. Then replace it with pea gravel or sand put down over a layer of wide Geotextile fabric. Make the layer of pea gravel or sand a minimum of 6" thick.
What kind of carp? Grass carp or common carp?
As for stocking the fish, while stocking might be warranted, I'd see what next year brings once you improve their bedding area (the sand/gravel area).
As for cover in the pond, look at the archives. There are many suggestions for cover, but the rule of thumb is 20%-25% of the pond surface area has to be cover for the fish. 1 acre pond? 1/4 acre of cover for the fish, split between dense cover for small fish and larger openings in the cover for predators to hang out. You will get bigger fish faster if you feed a good quality fish food like Optimal Bluegill food.
https://optimalfishfood.com/ Bluegill that are in good body shape will produce more eggs. If the fish are stunted, then it would be a good idea to stock new genetics, BUT you will need to thin the herd so to speak of what's in the pond now so the newly stocked fish don't become expensive fish food.
If you want big panfish, remove all the bass that are over 14" in length, and keep removing 15# of bass per year from the pond of all sizes.
In order of fixing your pond, here is what I'd do.
1) remove the muck, get gravel/sand in that area
2) add fish habitat to the pond
3) feed the fish to get them to grow quicker and get in better body condition.
As for aeration, just run the aerator during the winter when the pond ices over. Put the air station/diffuser head in 2' of water and leave it run so it melts about 10% of the ice on the pond. Running it at night is more important than running it during the day. If your pond doesn't get much wind during the summer, I'd run it in the deepest part of the pond at the very least from sunset to sunrise when the water temp is above 60°F, if not 24/7.
