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First, you need to meet Claire. Second, she needs to get settled in the East Wing. Third, you go to talk to her – she tells you about a tunnel to the (still locked) cafe. That opens up a few things on the eastern side of town.

Then, you make your way to the outskirts, down to the cave, and visit crystal dhena. That gives you the prompt about talking to Lyx about it. Lyx will visit you in the manor a couple times, but you'll get a clue the second time that leads you on a scavenger hunt that ultimately has you finding her. 

Once you do THAT, she's available (in principle) to start talking to about these leads, including the lead on the acid. She'll send you to the Shops, and there you go. The one with Kana at least completed for me; the others, though...