Is money paid to reinstate an employee to his previous position, through an influential person, considered a bribe, and does the money earned by the employee in this position become unlawful?

If the money will be paid to someone other than the manager, for them to use their influence or prestige to get you back your old job, then this is not considered bribery. Rather, it falls under the category of "payment for prestige." It is permissible to pay this money if you have a right to return to your previous job or if you were harmed by your transfer. The sin in this case is limited to the payment of the money, and the salary obtained is permissible unless there is another reason for its prohibition.