Answer: Melons can split if the weather is hot and they get too much water. The plants drink up the water, but the melon skins are thick and cannot expand quickly enough, so they split. You might try watering less frequently, but watering deeply.
The misshapen melons are usually caused by incomplete pollination. This can happen when the weather is really hot - pollen can become sterile when air temperatures are over 90F. So, even though insects are transferring pollen, it may be sterile and therefore the female flower is not completely pollenated. When you see these odd ball melons begining to grow, cut them from the plant so the plant can redirect its energy into producing more perfect fruit.
Best wishes with your melon patch.
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