走后门 To go through the back door. The Handian dictionary defines it as getting things done “through improper personal relationships or channels.”
托人情 To ask someone to put in a word on one’s behalf.
说情 To plead for another person, to ask for leniency or help on someone’s behalf. In the Qing novel The Scholars, frightened people beg a local gentleman for his card so that he will plead with the magistrate for them.