Is a man entitled to take the amount that the mechanic reduced from the repair cost by mutual agreement, and is this permissible in Islamic law?

The man is not permitted to take the amount the mechanic deducted from his fee. Rather, he must inform the car owner. If the owner grants him permission, then there is no objection. Otherwise, it belongs to the car owner, because an agent is not allowed to take any profit for himself without the permission of his principal. Instead, all profit belongs to the principal.