Live Listen disappear from control center sometimes on my iPad and iPhone

Re: "Why does Live Listen disappear from my control center sometimes and then I have to search for a new way to put it back? iPad mini and iPhone SE"

Am no expert in this area, and Apple is not here, as we are all Apple users, like yourself. Yet your request deserves some response, Seems there may be various possible reasons this situation may occur.

Eg: Keeping OS versions on each device up-to date with the newest compatible version may avoid issues.

(You can always check Settings > General > Software Updates, and back-up your data before updating.)

Headphones (or hearing aid, etc) may need to be connected in order to turn on / use / Live Listen

You may or may not have the option for Live Listen in the Accessibility Shortcut

(ie You can add it, if you wish:

Scroll down in Use Live Listen with Made for iPhone hearing aids - Apple Support for step-by-step guidance.)

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Above article also says: "Get help if you can’t see Live Listen

1. Make sure your hearing device is connected to your iPhone. You may need to pair or reconnect your hearing device.
2. Make sure your hearing device appears in the device selection area of the playback controls in Control Centre.
3. If you see a different accessory there, tap it and then select your hearing aid in the list that appears."

All the best :-)