If you are the custodial parent, only you can claim the credit, it can't be shared or transferred, even if you have a court order to allow your ex to claim the child tax credit.
You can only claim expenses that you actually paid. In the future, if your ex helps pay expenses, it might be helpful for your ex to pay you, and then you pay the care provider.
Of course, the maximum care expense you can use for the credit is $3000 for one child and $6000 for 2 or more children, so if you paid (for example) $4000 each for the care of one child, you already paid more than half, so adding more would not change the credit.
