Is it permissible to pay Zakat to the children of a man who was once well-off but was dismissed from his job and imprisoned, leaving the family with no source of income, and his wife is seeking employment?

If the family mentioned is in the state you described—that is, poor—then there is no objection to paying zakat (definition) to them, as they are among the poor who are eligible to receive zakat. Paying zakat to them is sufficient and fulfills the obligation.