Is my mother considered sinful if she paid a sum of money to a government employee to recover her delayed financial rights, knowing that she did not miss work and could not prove her right except through this method? And should she dispose of any excess if there is any?

Money paid to attain a right or repel an injustice is not considered a bribe. A bribe is what is paid to nullify a right or establish a falsehood. Therefore, if your mother has a right that she cannot obtain except by what she did, there is no sin upon her, and the sin is upon whoever compelled her to pay the money. However, she must not take more than her due, and if she takes an excess, she is obligated to return it.