It sort of sounds like “Together Again” by PinkTea, although not all the details match up. You take in a depressed girl, but she isn’t dropped off – you meet her when she’s about to kill herself and you convince her not to. The gameplay loop involves giving her a chore every day, and she’ll reject a lot of them until you build trust. It’s very heavy on BDSM themes.
However, I see that the page for the game has been quarantined on here now; not sure what’s going on.