Is it permissible to pay the prisoner's bail from Zakat before the completion of a year (hawl), and is he considered among those in debt (gharimin), if he embezzled public funds and was sentenced to prison, and he returned a portion of what he embezzled, which is all that he owns, and his family is left without a source of income?

For a debtor to be eligible for Zakat (definition), their debt must have been incurred for an act of obedience or a permissible matter. If it was for something unlawful, they should not be given Zakat unless they repent. So, if the debtor embezzled the money, they are not entitled to Zakat until they declare their repentance and mend their ways. As for paying Zakat before its due time, the preponderant opinion is that it is permissible for what requires a full year (hawl), such as livestock, money, and trade goods, provided the reason for its obligation (the complete nisab (definition)) is present. The proof for this is that the Prophet, peace and blessings be upon him, borrowed the Zakat of two years in advance from Al-Abbas. Based on this, there is no impediment to giving Zakat to the embezzler if they repent and mend their ways. If they do not repent, but their family and children are poor, it is permissible to give them from the Zakat to the extent of their need.