You might try planting wildflowers on some of the space, and then till them under end of the season. Wildflowers do not require good soil, but they will enrich the soil you have. I live on an old farm property, and I, too, have clay soil. I had to amend this for my beds and it has taken years, even with my horse and chicken manure, but you can also cut your lawn and add clippings, just watch to not add when the lawn has gone to seed, which would be maybe June, for you. ANY vegetation composted will help. DON'T GIVE UP!! We are all rooting for you!

Also, keep searching here for ways to garden cheap, and on the INET. Hobby gardeners are a cash cow for stores with seasonal garden centers and they want you to waste your money on plants that don't do well in your zone or soil, and then need to be replaced every year, like perennials that aren't zoned for your area. I once saw a white rose that was zoned for zone 10, for sale locally. Really?!?!?
I once met a woman at Rural King who was looking for annuals. She didn't think she knew much bc her MIL was a "Master Gardener" who spent $thousands every year to maintain her property!! I also ran into a real Master Gardener who is a visiting host of Mid American Gardener at the same store and SHE was looking for bargains there.
The easiest thing is to start from seed, although it is a time consuming winter activity. Usually my inside starts dry out on me. I try to bring in last year's annuals, most shade plants and I am always forgetting to water Them. One nice thing about gardening...they are only plants when they die, NOT kittens!