...Whatever happened in their past (if he even is her real father at all) must have strained their relationship...
Operating under the assumption that Lapis had a sister who had died due to reasons we are not yet (fully) aware of (which, by the way, is this the consensus? I'm uncertain what the general consensus is at the moment), I'm inclined to believe this dude, real father or otherwise, either has a direct relation to that or was in some sort of an antagonistic position against said sister. Perhaps he is that girl's biological father or something.
My read of him is that of an opportunistic slime. Something about the way he "showed interest" in Lapis felt beyond fake, and only to cynically get on her good side for whatever reason.
In any case, I love how this chapter develops our two mains -
Natori's flashback sheds some more light on her "background actor" nature and I'd say her general preparedness and skills and cements that it is truly part of her nature, being this benevolent goddess who somehow is also very generally prepared and eager to help.
And Lapis' monologue, a very self-hating monologue, I'd like to add, gives so much more volume to her as a character, and I really appreciate that. I'm also even more certain that Lapis isn't a villain by nature, she's playing the role of the bad guy because she feels like she has to, or something along those lines.
Also, I'm just a huge fan of the dynamic between these two. I'm really looking forward to seeing how things develop, there's so much potential, and so far the author has been doing a great job.