Cersei is sending Jaime to Riverrun to sort out the seige there, and he is not pleased about it On the journey he visits the Hayford's castle, whose son Tyrek disappeared in the Joffrey riots. Jaime is suspicious that Varys ought to have known those riots were likely but did not say anything. Interesting: perhaps we will learn of more games Varys was playing. Were he and Littlefinger ever seen in the same place?
Jaime practices fighting with his left hand with Ilyn Payne (because Payne can't talk and so can't tell people how bad Jaime is at fighting). I expect he'll get better off-screen and then surprise us all later by winning a fight.
Next Jaime meets Roger Hogg, another lord in a tower. I suppose there must be quite a lot of these minor lords living in draughty towers all over Westeros. I think I'd rather take being a commoner in a 21st century first world nation, thanks. At least no-one will turn up and start killing my sheep.
Jaime meets some "outlaws" and summarily hangs them and calls it justice. Nice.
Finally he ends up in Harrenhal and installs a new lord there. He also rescues Pia and Wylis Manderly who is only just alive. It's lucky he is because Cersei promised to send him to his dad as a reward for killing Davos a couple of chapters back. He also smacks Connington, Brienne's former betrothed, for insulting her, thus accidentally revealing that he cares about her.
