Is it permissible to count the amount that was paid to a zakat recipient a week before its due date—with the intention that it be zakat for the coming year—as due zakat?

If, when giving the money to the poor person, you intended it as your Zakat al-mal, then it is sufficient for you, as there is a consensus among all scholars regarding the permissibility of advancing Zakat for such a short period. However, if you did not intend it as Zakat, then it is not sufficient for you, due to the obligation of intention in Zakat, as the Prophet (peace and blessings be upon him) said: "Deeds are only by intentions, and every person will have what he intended."