@fruitscandy It was explained deeper in the novel, but the sex was kind of a promise that Rakan would redeem her.
As a top courtesan, Rakan couldn't afford her so she allowed him to get her pregnant, hence lowering her price. However, Rakan went to war(?) and couldn't redeem her. Pregnant courtesans have very low value, so FS had to accept lowly customers to sleep with. Rokushokan also lost prestige and almost became bankrupt, since their top courtesan was ruined. FS sent her pinky (and tried to cut maomao's too) to remind Rakan that he broke his promise. Pinky swear (yubikiri) is a big symbol of promise in japan.
Now, rakan's fault in this is that he didn't think of the consequences of getting FS pregnant and leaving. He thought he can just come back and then redeem her upon his return, when it was URGENT that he did so bec FS reputation is already damaged.