Yes, if you are married filing separate, the spouse who claims the child as a dependent will also claim all the other tax benefits associated with the child. These benefits include:
Child tax credit
Exclusion from income for dependent or dependents care benefits, but the maximum amount is half the normal allowed.
However, you can each claim the medical expenses that you paid for your child, if you're itemizing deductions.
Note: The credit for child and dependent care expenses is normally not available for MFS returns. And the EIC is never available for MFS returns.You and your spouse can't agree to splitting benefits between yourselves.