> I feel like I'm able to correlate "better sleep" shown in the app with "better mood" or "feeling better" the day after

If you look at the app and see that it says "better sleep" before starting the day and then notice that you had a good mood or felt better after that could be a placebo effect.

You could check that by not looking at the app until after your day is done and you have noted if you had a good mood or felt better. If the app and your day's mood still correlate then it probably is something real.

On the other hand checking if it is real risks making it no longer work, and a good mood is a good mood even if it was caused by a placebo. I'm not sure I'd check.