Let's see what I can remember:
Basically, this month was just a conversation between the weak and bedridden Manji and Master Sori.  Master Sori was filling him in on what was going on around him.  He also told Manji that he'd be better off not trusting Mugai-ryu (and Hyakurin).  As it turns out, Mugai-ryu is a collection of criminals that were supposed to be executed by the government.  Instead, though, the government allowed them to live if they worked for them by entering this Mugai-ryu group and did whatever the government ordered them to.  In other words, all the Mugai-ryu chaps probably did some heinous stuff to get where they are today.  I guess that explains Shira a little better.

But that leaves us with a question- why did the Bakufu hire Anotsu and Ittoryu to train the government soldiers, yet at the same time are trying to kill him through Mugai-ryu?  Plots within plots.

Anyway, Master Sori goes on to say that he knows Hyakurin, or at least knew of her.  Her husband was a friend of his, and a powerful samurai.  Hyakurin killed him.  Sori doesn't know why she killed him, only that she definately did do it.

So that's basically why Mugai-ryu and Hyakurin sent Manji out to kill a couple of Itto-ryu guys.  Maybe they wanted him to take out some Itto-ryu boys along the way, that is uncertain.  But these Mugai-ryu guys are a bunch of criminals and assasins and thieves.  So it didn't occour to them to send Manji out to Master Sori or another guy with authoritative power to get a passport.  Their first thought was that the only way to accomplish the mission had to be through violence or thievery.  Sori tells Manji that if he came by he could have given him the passport in a second.  Manji reacts to that rather... colorfully.

And Rin is on her way, following Anotsu's trail.