A clay back yard

If you want plants or anything, I would go with pots or raised gardens to avoid the clay soil. You could put them where you have the most problem with cracking or if there is an area that the grass won't grow.

water the yard and the cracks

We have a bag on back of our lawn tractor that picks up grass when we cut grass in the summer. We have clay soil also. Quite a few years of throwing grass clippings into our garden was a lot of work. And it helped. Lately. every year now when we plow our garden, I save card board over the winter. And we use this to keep the weeds down. The weeds do come up sometimes, put we throw some of the grass over the card board and let it deteriate over the winter from the sum, heat, snow and rain. Plow year after year and every thing goes into the soil so it is not as clumpy. I have fountain grass growing in my yard, We cut them down every fall and lay them in our garden. They deteriate also. Good luck.

the only thing you can do is to start to amend the soil...without plowing everything up and adding topsoil, you could topdress with topsoil each spring, mulch your grass clippings rather than raking, throw some peat moss down in the fall....this will not be an overnight solution but over the course of a few seasons should help without spending a fortune on water etc.

Your lawn is cracking because the clay soil is drying out. It needs to be watered consistently and well. The second thing to do is start a compost pile. Put all grass clippings, raked leaves, etc. in it. give it some water once in a while to keep it slightly damp, turn the pile over if you want or not. Next year you will have great black compost for your garden or scatter it over the lawn. Keep doing this consistently and you will change the clay to good soil. It takes a long time, but does not cost anything.

Try mixing in some sand as well as compost