".... let partner claim them as dependents and file HoH using those children."

Who did the kids live with?

The custodial parent would use the kids as qualifiers for HOH/EIC/Child care.  The non custodial parent would get the RRC/CTC for those kids.

The non custodial parent shouldn't have filed as HOH.

Or did they all live together?  if so, they cant split up the credits, one person takes all.