Mat does indeed use the spot on the wall that Rand found, and while sneaking around he gets some Exact Eavesdropping (or as I call it, the Jullian Fellowes Special) from Gaebril and his underling Comar, who is assigned to kill Elayne. Then he’s caught by Tallanvor, who is luckily one of the original guards. He finally gets to give Morgase the letter, and she’s happy enough that Elayne is still alive that she’s put in a good enough mood not to throw him in jail. But Gaebril is also there, so he doesn’t get to tell her anything about that, and is just escorted back out.
My guess for now: Gaebril is actually Bors/Carridin. His turning Morgase against the Aes Sedai combined with talking about putting people to the question definitely indicate a Whitecloak. At the same time, Tallanvor is clearly being set up as Mat’s ally inside the palace, and may even get a promotion to viewpoint character if he survives this book.