Since you already used the child's ssn for the credit and shouldn't have, then the mother of the child who was entitled to claim the child for that purpose will have to file a paper return to claim the child.
Once a social security number of a taxpayer or dependent has been used in the system it cannot be used a second time for the same tax year.
It will take longer to process the paper return for her refund if she includes the child's SSN in it, but that is one way she can file it and get what she is entitled to receive.
The only other way is to file her return leaving the child off and then, once she gets that return processed, she can file an amended return including the child on it and it will be a paper return, filing for the additional refund.
Even though you will be amending your return to remove the child, the child's ssn still cannot be used for electronic filing for the tax year because it was already used. Amending the return does not fix that.
