That whole scenario would be an interesting project for me. I would start by pumping the water out of the supposed spring area and see if and how fast it replenishes, you may be able to pipe it from a little reservoir off to the side to be able to redo the pond area, deepen it and clean out the muck, then divert the water back into the pond and have the potential of a spring fed pond.
I agree with QA that the leak by the tree stump looks like a rotted tree root or even an animal burrow that burrowed under the stump using the trees roots as support for the roof of their tunnel, which is a very common thing for burrowing animals to do.
