On xperia 10 III, the camera response can be very slow, > 3 seconds, sometimes even 10 seconds. I noticed that it is connected with the process of finding focus. Under complicated light situations (like moving lights), it is impossible to take a photo.
I think (anyway!) a good option in general would be to be manual focus (select and keep): set a focus point and then be able to keep it, and then being able to just shoot a photo. I find it a bit earger that the camera is dictating how much focus the photo should have. As a side effect, this would at least help a bit with the camera on the 10-iii.