A pond's goal is to fill itself back in with dirt. Our goal is to prevent that. There are countless options and ways to accomplish your goals.
First determine what is causing your muddy water by putting some in a jar and letting it set for a while. This will determine if the turbidity is from suspended solids or if something is mixing up the water. Givin your tree cover to prevent wind/wave action, I suspect Bullhead Catfish, Common Carp, or both.
Second, clearing out a lot of the trees will allow more sunlight to reach the water for more effective bacteria development to speed the decay of the detritus, as would aeration. This will also lessen the volume of dead leaves.
Frankly, if the budget allows, a total makeover of your pond would probably be the least expensive route to take. It will allow you to complete everything at once and allow you to re-stock to match your goals.
