Okay, I think I was way off. Here's from an online Farmer's Almanac:

29th-30th Extra Good For Planting Fall Lettuce, Cabbage, Cauliflower, Collards, And Leafy Vegetables. All Aboveground Crops Planted Now Will Do Well. Plant Seedbeds.

From Pegi Ballister-Howells' "Month by Month Gardening in New Jersey", the following can be direct seeded in

July:
cabbage, cauliflower, brussels sprouts, greenbeans, peas, summer squash, sweet corn.

August:
beets, leafy greens, lettuces, chinese cabbage

Radishes and cukes are listed for June, but you could given them a try, depending on how much shade and how dense - cukes sometimes appreciate a little shade from the hottest summer sun.

Again, depending on the amount and density of shade, green beans may be worth a try. Zucchini is so vigorous, maybe give it a try, especially a bush type. Expect it to spread more in the shade.

It will be a good experiment and learning experience for the children as long as they have some successes in there. Besides, it's always fun while gardening to test the limits and the investment in seeds is pretty low.

Your cukes, zucchini and certain other veggies might be more susceptible to powdery mildew. Be especially careful about garden hygiene, water the soil not the plants, and be on the lookout for early symptoms - there are some kid and environmentally friendly options to control p.m.

I also have only a few small spots that get half-day sun, but I've just got to have my veggie garden! I've even planted some in the front yard, much to my neighbor's dismay ... she's not up on the "edible landscaping" trend yet.