Is it permissible to send a sum of money to a former teacher indirectly (such as by recharging a phone balance) without informing or embarrassing him, in order to repay an outstanding amount that he did not mention?

With repentance, it is sufficient to deliver the amount to its owner, even if indirectly, such as via a postal transfer. It is not sufficient to transfer it to his phone balance or his children's accounts unless they commit to delivering it and informing him that it is a debt from the student.