Does the aforementioned family, whose members are extravagant in spending while the head of the family is poor and ill, deserve zakat, and must zakat be paid again if they are not eligible?

If the family's provider's income is insufficient for their needs, they are poor and it is permissible to give them zakat (definition). What they need in terms of money, beyond the father's income for an entire year, should be estimated and given to them. It is not permissible to give them more than their needs. If the zakat payer fears that the wife will mismanage the money, then the money must reach the husband directly. Then, he should advise and inform him about the mismanagement by his wife and children. If this does not help, the zakat payer can identify their needs and provide them from zakat money as an agent on their behalf. The zakat payer must verify that the recipient is poor and eligible for zakat. If it is proven that he is not eligible after the payment, the payer's liability is cleared, and the sin is upon the recipient.