Is the remaining amount for the lesson's fee, which the student was unable to pay the teacher due to the latter's absence, considered a debt upon the student, and is it permissible to give it out as charity, or must it be paid to the teacher?

You are obligated to pay for the lessons the teacher taught you, unless he forgives you, due to the generality of Allah's statement: "O you who have believed, fulfill [all] contracts." Your responsibility is not discharged by donating the price as charity, unless you are unable to deliver it to him.