Window cranks disengages when trying to open window

Ouch. The "worm gear" that fits into the hole in your first photo is not meant to come out. Those things get pressed into the assembly before they leave the factory, and they cannot be repaired. You'll have to swap out the whole operator.

Now, I'm confident that we can find a match, but we'll need to see the operator out of the window, so that we can see the gears and arms exposed. Can you send us a photo? You may also want to check out the video on the right side of the page, "How to Replace a Casement Window Operator."