I had this problem! In between emails with the people at Sleep Cycle, it magically started working for me. I say "magical" because since I was trying so many different things to troubleshoot, I don't know exactly what fixed it.
Luckily, I had a second iPhone X that I could troubleshoot on. Here's what I did...at some point in this mess, the Sleep Cycle app started working again:
I used Decipher Repair (which is just software to "fix" corrupt iPhone backups), a few different times on different backups. Every time I would change the parameters of the backup repair and then sync it to the new phone. After each try, I would factory reset the phone and try something else. The last thing I tried was to delete any app that I don't use on a regular basis, create an iTunes backup of my phone, run Decipher Repair twice (to fix any potential corruption), restore the phone to the "fixed" backup, then when launching Sleep Cycle for the first time I DENIED microphone access, started the app, ended it, then went to settings and toggled on "microphone detection" in the app, which forced me to go into system settings and allow access to the microphone there. Once doing all that it magically started working.
The people at Sleep Cycle suggested this:
"Ah, the plot thickens. To me this sounds like Apple has made a mistake in the way permissions are "moved" between your iCloud account to different devices, specifically going to an X.
As you know you need to allow Sleep Cycle (and Voice memo) to have permissions to read the microphone.
These settings are stored within your iCloud account, and should be moved between devices. It sounds like, when moving to an X, the permissions is moved as well but from a technical perspective it's not working. It's set to allow in the OS, but somehow, the setting "behind the UI" is probably off.
So, if you set your X up clean, my guess is that both Voice memo and Sleep Cycle will work fine. As soon as you restore your backup though, you're screwed, because then it starts to mess around with your permissions.
Could you try turning off the microphone permissions and then set it to allow once again? Perhaps this 'resets' the behaviour somehow. You're not alone with this issue, it's already popping up in forums - and I've reached out to Apple about it too so they're aware of the iOS bug."
Anyway, I know I haven't offered a clear solution...but maybe somewhere in there you can find something to help.
