I'm in the UK so the weather is different, but I would say you went for the difficult options - probably by accident.
Cucumbers and watermelons need good soil and lots of water. If you only grow a small number of them (considering the small size of your plot), also you have to get lucky with fertilizing the flowers, since there are separate male and female flowers and they both have to be in flower at the same time, and visited by the same insect! Or for some varieties, you need to remove the male flowers, because pollination and setting seed affects the taste of the crop (and not in a good way!) If you want to try again, plant them in grow-bags not direct in the ground. At least that will eliminate any problems with poor soil and pest and diseases living in the soil.
If you want to grow brassicas, broccoli and cauliflowers are more difficult than "plain" cabbage. They are harder to grow because the plant has to fully develop, without "bolting" (flowering and setting seed) otherwise you don't get a crop. If a cabbage plant grows poorly, at least you can eat it even though it's small.
I can't really comment on corn. In most of the UK (including where I live) the unpredictable weather makes it not worth even trying, but of course it is native to the USA. But as with cucumbers and watermelons, you have to grow a big enough area of corn so the flowers will get pollinated by the wind, otherwise you won't will only be growing biofuel, not food!
I wouldn't bother with potatoes. Your plot is too small to grow a meaningful amount (unless you only want to grow one saucepan full), and if you did produce more you would need the right conditions (cool and dark) to store them. I certainly wouldn't cover some of the garden with wood chips which will take years to decompose just to grow a few.
I would suggest you go for quick growing plants (any sort of salad vegetable), plus peas and beans. Pole and dwarf (or French) beans will both produce huge crops for a long season if you pick them regularly so they keep flowering. Peas are not so productive, but the taste you get from shelling and eating peas (raw) within minutes of picking them is something you can't buy from a shop.
Tomatoes (again, grown in growbags not in the ground) would be another option, if you have the time to water them as much as they need it (which may be more than once a day).