Crops to Grow with Limited Sun

I started a vegetable garden about 3 years ago that now has one partially shaded area. I have tried growing onions, beets and cucumbers without success. What will do well in only a couple hours of sun?

Leafy crops like lettuce, spinach and greens are the most shade tolerant. Root crops such as beets and turnips need more sunlight than leafy vegetables but will tolerate partial shade. Vegetables such as tomatoes, peppers, broccoli, cauliflower, melons, squash and other vegetable that we eat the flowers or fruit of the plant, need the most sunlight. These vegetables grow and produce best in full sun.