engineers the discovery of horrifying information that leads to Marten losing his job.
Maybe Tai doesn't know about the Bangin' trustees but told Claire earlier about how Marten got a mysterious promotion; Claire confronts the trustees and they tell her why since she introduces herself as his partner (and they think it's adorable they're together [they're not the type to punish them for an office romance]).
She uses against him by convincing the trustees to fire Marten, since she doesn't think it'll be good for her education if she's distracted at her internship. The trustees are taken aback (since they thought being okay with Marten and Claires relationship would get them Claire's discretion in return) but they try to maintain a business outlook and agree that Marten didn't earn that raise really and they could take it away from him with nearly no repercussions if they remind him they'll be equally discreet on his relationship with an intern (if the trustees relationship is spilled by Marten they will counter-accuse him of abusing his position to enter a relationship with an intern).
Claire leaves and the trustees call in Tai, whom they admit their relationship and how Marten got his raise. Tai brings up a raise for herself, and explains it's not that Marten got the raise it's that Tai knew before Marten did (they've done this before) but didn't get caught when walking in on them [She was going there to use the copier, not leading a partner] and never got her own chance for a raise. The Trustees of the Library agree, she can have her raise... if she fires Marten.
Tai says she has to think about it, but the trustees are understandably wary and ask her if she just needs to call someone to ask advice, if she can do that can she stick around at their house? Tai agrees, knowing that their tryst could ruin their positions as trustees (or... whatever their title was I keep calling them trustees). and calls Dora. Tai gets some privacy by the other two leaving the room, and tells Dora the story of how Marten got his raise, how Claire made an ultimatum about his job because of her jealousy of Faye, and she had been totally lying to the trustees when she said she knew about them before Marten did for a raise.
Kay I'm going to read the rest of your comment cause I got so hung up on the quoted part.