Do NOT provide care in June without FULL advanced payment. If she is terminating care then she pays until this Friday and the child do NOT come tomorrow or Friday. Contract is over, she is breaking it. She pays but care is done.

September is NOT an option. She has broken the contract and has shown she doesn't not care to abide by the contract...you cannot let her resume care in a few months to break the contract again when she decides to.

Call her tonight. Let her know that he is not welcome tomorrow unless she has payment for June in hand (not post dated). Let her know her space so no longer available for September that you will fill it with a family that can abide by the signed contract.

I don't know that I would go through much trouble to force the month notice payment (as per contract). If she legitimately doesn't have it then it will be a lot of time and effort and lost money on your part to get it. You might be best to find a new family and fill the spot ASAP. It should be easier to fill the spot long term than just for a few months. If having trouble filling it, then fill it for the summer while looking for a family to start in the fall.