Are we going to end up having a physical relationship again and is this how it is always going to be? 1 month of being together followed by a period of friendship before she decides she is leaving him again and re-embarking in a relationship with me? Should I hang in there patiently in case there is a chance of us being together or should I cut all contact?
Here's an idea, why don't you actually take responsibility for your own actions?
Literally all those questions can be answered by you:
Are you going to have sex again?
Is this how its always going to be?
Should you wait and hope?
Should you cut off contact?
Make a decision about what your answer is to those questions and then act accordingly.
Waiting for her to "decide" and going along with whatever happens next is you absolutely copping out of being a adult and having accountability for your actions.
Personally I wouldn't want to shag someone who was behaving so cruelly towards her kids. The cheating and yo-yo-ing is bad enough but being such a dick to her kids takes the biscuit.
If you're happy to carry on like this forever, you're not a very nice person and also a mug.
If you wouldn't be happy to carry on like this forever then cut contact. If she doesn't leave her partner for you then that's you answer.
She wants you but only if it doesn't rock the boat at home too much.
Be an adult, make your own choices.