No. Payments received by a state agency are counted as the child's total support but are not counted as support provided by the foster parent or the child.
Two other important things to note:
(1) Payments received from state or local governments are generally not included in the taxpayer's income.
(2) An exemption may be claimed for a foster child only if the child is a qualifying child or a qualifying relative. (See the link below)
Qualifying Child and Qualifying Relative
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