Possibly Purslane, a plant that grows prolifically from seed and any bits and pieces of the plant left behind when cleaning up. This is a summer annual that grows best in soils a bit less than a good garden soil although it has adapted well to good, rich garden soils.

You could till it in, add what soil amendments are necessary to grow those melons and then lay down good heavy mulches which can keep the stuff from growing. I have covered Purslane, Chickweed, and etc. with newspaper and cardboard and did not have more of this to contend with until the following growing season.

kimmq is kimmsr.

I agree with kimmq, purslane. It's a pain in the neck to deal with, but not horrible. Mulch is the best way to keep it under control, as unless you get the main tuber, it will keep coming back.

Purslane (Portulaca oleracea) is an annual, full sun plant. Also edible. If you're going to grow melons, I wouldn't worry about purslane. The shade from the large leaves of the melon vines will probably get rid of it. If not, the purslane stays low to the ground, keeping it shaded and less parched from the baking sun.

The larger foliage looks like it may be Canada thistle. If so, removing the taproot of each one is necessary. A dandelion fork should be able to pop them out when the ground is moist and soft.