Our back yard—which is on a slope towards the back that sheds the surface water off our yard—is also heavy on clay soil and would make the spongy sound when you walked on it a day or two after a rain shower.

Once we started having it aerated every fall, that reduced most of the spongy sound since the water got deeper into the ground and more loose leaf/grass cuttings got into it.

That might be the cheapest/fastest route you could take. If you’re physically able to do it yourself, rent an aerator from HD and give it a pass. Otherwise, a lawn maintenance person/company can do this for you for a fee.

Regardless of who does it, make sure to walk around your yard and mark any solid things on or near the surface (exposed roots, trunks, sprinklers, rocks, etc.) in the ground that you’re aware of otherwise the aerator will buck on you if it hits it.