Clay can be very good soil for growing in, if it has sufficient organic matter. Most companies that sell 'topsoil' don't have a very good organic content.
I grow under very different circumstances than you, but the very first thing I would do is aerate. I prefer doing this by hand with a garden fork, but there are machines that remove plugs and leave them on top of the soil. After that I'd spread at least 1/4 an inch of manure across the whole yard. I trust that in addition to mowing high, you are already leaving all your trimmings to nourish the soil life.
I successfully brought plants back into the barren (not even weeds) third of my front yard with just aerating and lightly scattering some excess wood chips that I was working with. Manure or
compost would have been even better. Every year the ratio of grass to mixed greens shifts further to the grass side.