Will concrete poured over wet ground be more likely to crack and disintegrate?
Civil engineer here.  The concrete needs to stay wet in order to cure properly. As noted before, concrete doesn't dry out to harden, it cures through a chemical reaction that needs water to facilitate the reaction. If the ground is dry, the ground will absorb moisture from the concrete and it won't cure properly.  The ground should be very moist and compacted as well as you can manage.  

Also, wet clay is notoriously weak and you may end up with uneven support under the concrete, which will then lead to it cracking and settling after it's cured.  Hard to say without seeing it but if you can remove as much clay from beneath the concrete slab, you'll have less chance of it settling and cracking.  This is especially true if the clay is expansive.  It will expand when wet and shrink when dry and your concrete slab will be a disaster. Best to get as much clay removed as possible, compact the sub-grade evenly, and make sure to water the ground before placing the concrete.  

Have you thought about proper joints?  Concrete will crack.  If you can get it to crack along a joint, no one will see it. Look at some professionally done concrete flatwork and you'll see what I mean.  Good luck.