Is it Islamically permissible to seek mediation for acceptance into a government job, knowing that employment in those institutions is only achieved through mediation?

If the debates do not involve anything unlawful, and you are qualified to participate and need an intercessor to be accepted, then there is no harm in seeking someone to intercede for you, provided that this intercession does not result in depriving others of their right. However, if the intercession leads to depriving someone who is more deserving, or causes undue hardship to others, then it is forbidden.