Since the scholarship income exceeds the cost of room and board plus tuition and related expenses, it becomes income to your child. Add the $1300 to your child's wages and it is less than the amount required to file. Your child can file a return to claim any tax withheld, otherwise, no need to file a return.
It is also considered unearned income for the kiddie tax, you are below the limits for filing that income on your return.
You are not being taxed on the $1300 at all unless you have some other circumstances not disclosed.
References:
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