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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...

I'm rather stuck in Act 3 and unsure how to proceed. I'm at a point where multiple leads are like "talk to Lyx" but when I do she has no options for me. I 1) have the hydrochloric acid, and 2) have introduced Lyx to Viola for the flying bar, and 3) have completed the dream sequence for the nexus, and 4) am trying to find lomeg, but it is not in any of the available eastern districts of town. Reached Emma's power cap before I find someone else to talk to. Saved Kana and had her punch her mom. I "explored" all the various locations at each time of the day. Jessica at the Ruined Farm doesn't offer any more dialogue.

It really is a pickle, and there're no clear walkthrough or guides for this build. Folks seem to be getting past this point so I dunno what I'm missing.