Is the additional amount distributed to the poor counted as part of the Zakat on wealth, knowing that the Zakat on the giver's wealth has already become due?

It is not permissible to count the money distributed in excess of what the client gave you from the zakat, because intention is a condition for the validity of zakat. The proof for this is the saying of the Prophet, peace and blessings be upon him: "Indeed, actions are by intentions." So, if you intended an absolute charity, it would not be valid, even if you donated all your money. What is disbursed from your wealth and owned by the poor will be in the balance of your good deeds, God willing, and will be counted as charity.