Is it permissible to pay off a debt incurred by a relative – who took on this debt to travel in search of work but was unsuccessful – from the zakat funds allocated for the poor, which I have been entrusted to distribute?

The fundamental principle is that the agent acts within the bounds of what the principal has authorized. If the zakat (definition) payer has specified that it be spent on the needy (masaakeen), then it is not permissible to disburse it to a debtor or anyone else. However, if there is an indication that the payer's intention in specifying the needy was not to restrict it to them, but rather to ensure the zakat reaches its rightful recipients, then it is permissible to give it to any of its designated categories, such as a debtor, provided that the debt was not incurred as a result of disobedience.