What a fantastic idea to start with a cover crop to start improving your soil health. Some cover crop plants that are commonly used in the summer are: Buckwheat, soybeans, black-eyed-peas or cow peas. Cover crops are more commonly used in a farm field setting, so you'll want to be sure to cut your cover crop down before the plants go to seed. If they go to seed you might introduce an unintended weed problem to your garden beds. 

Here's some information on using cover crops in the garden:

https://barron.extension.wisc.edu/files/2023/02/Cover-Crops-for-the-Home-Garden-1.pdf

I would also recommend that you start with a soil test. 

https://homesoiltest.msu.edu/

This will tell you the nutrition that might need to be added to your soil for healthy plant growth. It will also tell you how much organic matter you have and how much you could add. Incorporating the killed cover crop is one way to add organic matter, but you can also add organic matter by applying compost and using natural mulch materials like bark mulch, straw or pine needles. More resources on this below.

https://www.canr.msu.edu/resources/dont_guess_soil_test_get_your_home_lawn_and_garden_soil_test_kit_today

https://www.canr.msu.edu/resources/smart_gardens_begin_with_healthy_soil

https://www.canr.msu.edu/resources/smart_gardeners_improve_their_soil_and_weed_control_with_organic_mulch