Bush beans are easy to plant & will improve the soil because they're a legume. You can even pick some of the beans to eat. I've used them planted 9 to a square foot. They don't mind if the soil bed isn't smooth or has rocks. You can stomp the dying vines down at the end of the season & wait until spring to turn them under.

A combination of buckwheat and calendula around the outer edges of the beans will flower & attract the beneficial insects to your garden. Calendula germinates now & grows quickly.

All of those will need summer water unless you mulch well & we have some rain in July & August.

You can probably get the seeds at a feed store cheaper rather than a garden store.

I'd also suggest a combination of compost ingredients layered over the top of the soil where you're not planting a cover crop or once it's done in fall. Collect leaves & herbicide grass clippings and just spread layers of grass clippings no more than 4" thick. Plus if you can get some manure add that to your sheet mulch & your soil will be a lot easier to dig.

I had poor, rocky glatial till soil next to my driveway where we have the most sunshine and over time transformed it into soft hand diggable soil + 100s of buckets of rocks. What made the most difference wasn't a layer of chicken or rabbit manure thinly applied in the fall & winter over 10 years though that helped the soil. It was the 12" or so of horse manure + the bedding mounded on permanent beds in fall for several years. Quite a big difference now in the percentage of organic matter. Our backyard flock and rabbits just didn't have the volume our soil needed. An inch or two of manure and bedding just disappears even with the potent chicken and rabbit manures.

The best compliment I received was a question:
Where did you get all that nice topsoil?

My reply:
We made it.

Maybe you want to put in a gate to that area so you can unload easily into a cart from the pick up & dump. We use a trailer made from an old pick up to haul manure & it works great plus doesn't have to be spic & span when we're done.