Answer: Annual ryegrass is a good choice for between the rows. It grows quickly, tolerates mowing, grows thickly to outcompete weeds, and contributes a lot of organic matter to soil. It's also tolerant of a variety of soil conditions. Sow in spring at a rate of 1-2 lbs/1000 sq ft. Soybeans also tolerate poor drainage, and fix nitrogen in the soil. You may have to search a bit to find a soybean that doesn't shade out your crops though. If you purchase soybean seed that is not "inoculated" with beneficial, nitrogen-fixing bacteria, purchase some of this inoculum and use as directed.
Start them when the soil warms and risk of frost has passed. You can sow seed for cover crops by simply "broadcasting" them over the smoothly raked seedbed. Mixing seed with fine sand helps distribute it evenly. Then cover the seed. For rye, you need only rake the soil to make sure the seed makes contact. Soybeans should have 1/4-1/2" of soil or compost coverage. Sprinkle the area to make sure there's enough moisture for germination.
If you've never gardened in clay soil before, take care not to work in the garden when the soil is wet. To do so is to risk turning your garden into a hard, tennis court surface!
There is also the option of growing your crops so intensively that they shade out competing weeds. You can read more about this growing method in John Jeavons' book, How to Grow More Vegetables...", Ten Speed Press, ISBN: 0898157676.
Best of luck!
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