Is the official's action of providing financial aid that does not appear as support, then asking the manager to handle it "with anything so it doesn't cause problems," followed by the manager filling out a form with the amount as "service rendered," a mistake? And who is responsible for that?

As long as the manager asks the employee to conceal his action, this indicates that he does not have the authority to dispose of this money in that manner. Therefore, it is not permissible for the employee to assist him in this prohibited act, for Allah, the Almighty, says: "And do not cooperate in sin and aggression." The pleasure of Allah is more worthy than the pleasure of creation, and His displeasure is more worthy of avoidance than the displeasure of creation. So, whoever seeks Allah's pleasure by incurring the displeasure of people, Allah will suffice him against the burden of people. And whoever seeks the pleasure of people by incurring Allah's displeasure, Allah will entrust him to people.