I'm going to assume that pic is where the water line tile will go? Is that material just below coping new from the coping install?

Actually the pool never had a waterline tile just concrete up to the coping. And yes the material that is cracking is the concrete that was holding the bullnose coping on the pool.

Because the material under the coping keeps cracking we are not trying to understand how to fix the problem and the coping properly.

The coping stones should have a mortar joint between the stones to allow for expansion and contraction. They are moving as a "unit" over the winter, and causing the setting material to crack away beneath. the stones need to move individually.

There should also a minimum of a 1/2" joint between the back of the coping stone and your concrete patio.The joint gets filled with self leveling caulk type material such as Deck-O-Seal.

If the patio is up against or very close to close the back of the coping, it will move over the winter and push the coping around and cause cracks. If you had tile in place, that probably would have cracked as well.

An 8" beam does not provide a stable base for 12" material. With a 2" overhang into the pool, there's still 2" hanging off the back which subject to frost heaves.

Thank you so much for answering and providing feedback. This has been very frustrating for me to deal with and understand what the real issue is. It sounds like we need to remove all the coping and replace it with one that actually fits the pool beam. Followed by a joint that will fit between the patio stone deck and the coping.

Is polymer sand considered a substitute for for joint material instead of Deck-O-Seal?

It's a completely different material.

The polymer sand is ok. Key thing is to maintain a joint.

Update. I spoke with the company that installed the coping and they agreed that the coping is not the right size for the pool beam and that it needs to be cut. The company is suggesting that they will cut the coping without taking the stone off the pool. Is this something that can be done? Can we also install concrete between the coping stones that sit on the pool beam without taking them off? Also I don’t have a pool deck on one side of the pool it’s just grass, is this okay? Or should there be some sort of flex joint between the grass and the pool?