What should I do with the money I received for lessons I was unable to give to a student? Is the student's father's forgiveness of me valid? Is it permissible for me to offer other assistance equivalent to the value of this money?

The financial right belongs to the entity paying the wage (the state), but the questioner is entitled to the wage simply by presenting himself for work during its appointed time, even if he does not perform any work, and there is no fault upon him.