Is it permissible to count the amount paid to a needy female relative as part of zakat on money, even though the intention at the time of payment was to help and not for zakat, and does this suffice from giving out new zakat?

The intention must be present when paying money for it to be considered Zakat (definition), and the money must be given to those who are eligible, from among the eight categories of Zakat recipients. If there was no intention to pay Zakat, then the amount paid will not be counted as part of the obligatory Zakat, and the Zakat remains a debt that must be paid as soon as one is able. Paying Zakat is an immediate obligation once its time is due, and it is not permissible to delay it except for a valid excuse. Forgetfulness is excused for the Ummah, but whenever the forgetful person remembers, he must pay the Zakat immediately.