Q.Mysterious melon vine crawling along my backyard
What isi it? All I’ve done is periodically place compost in this spot. Suddenly large clover shaped green leaves with tiny yellow flowers, and what look like honeydews are growing on the vine!????? What are they?????

The problem with all squash, melons and cucumbers is that they freely hybridize. That means if you did not plant them, they may be a cross between two plants from last year. Did you plant any squash, melons or cucumbers then?
If you threw seeds from squash you purchased into your compost pile, you may have that variety with no hybridizing.
The best way to tell what you have it to cut them open and see what's inside, what it smells like, what it tastes like...