If you have an established lawn, regular fertilizer and watering should keep it in good condition.  This web page describing calcium sulfate seem to hint at its use as a fertilizer, but nothing clearly states that it will break up clay soil. From this web site: 
/agbmps.osu.edu/bmp/amending-soils-lime-or-gypsum-nrcs-333

A soil test from the U of MN is a better idea. Go to:  "soiltest.cfans.umn.edu" for submission directions. The test should cost $19.00. 

This web page describes lawn care. "extension.umn.edu/lawns-and-landscapes/lawn-care."

---

I've requested that your question be reassigned to someone else with more turf knowledge. 

---

Core aeration will help. It pushes tines into the soil and removes thin  tubes of dirt. Water and organic matter (like grass clippings) can then penetrate lower into the soil. It does not improve it very deeply, but gets the layer that supports the grass roots. The yard looks a little funky for a week or so after aerating until the plugs start to break down, but it helps with compacted soil. 

You can hire a company to do it or rent a machine that looks kind of like a lawn mower. I live in a rural area and also have clay. It gets compressed from equipment (ATV, backhoe when putting in the dock, etc.) so I bought something that tows behind the ATV or riding mower. After a few years of that, my yard looks a lot better and the grass has a fighting chance over the weeds. 

Don't fall for the push-in things like the spikes that go on shoes. They don't work. Well, they are probably good for exercise, but they don't do anything for your yard. 

See the section here on core aeration: 
https://extension.umn.edu/lawn-care/how-control-thatch-your-lawn#:~:text=The%20best%20time%20to%20aerate,the%20lawn%20twice%20before%20aerifying.