Is it permissible to count a sum of money given to a poor person as part of zakat, and to renew the intention for zakat after the payment, knowing that the intention for zakat or charity was not recalled at the time of giving?

Intention (niyyah) is a prerequisite for the validity of zakat. If a person doubts whether he intended his payment as zakat or a voluntary charity (sadaqat tatawwu'), the original presumption is that the intention for zakat did not occur. It is not valid to intend the money that was disbursed as zakat after it has been paid, because the intention would be delayed past the time of payment.