What is the ruling on giving the school bus supervisor a monthly sum to ensure the child is not forgotten, and is this considered a bribe?

It is not permissible to give the school bus driver or supervisor a gift or a wage for the work that is obligatory upon him, because he receives a salary from his employer. Gifts to workers are considered an illicit gain (ghulul) and are a form of prohibited bribery, and they may lead to your children being favored over others.