Button has started to produce a mechanical sounding click

It might feel like a physical click, but it isn't (which you can confirm by trying with the phone switched off. (I'm assuming you are talking about the home button here?

Go to settings>general>home button, where you can change the intensity - but not turn it off altogether.

Hi

First thing to try is a forced restart:

* press and quickly release the Volume Up button.

* Press and quickly release the Volume Down button.

* Then press and hold the Side button until you see the Apple logo. (KEEP it pressed until the apple Logo appears)

Go o settings>Sounds & haptics>keyboard click - turn off.

ahh. I would make an appointment at your nearest Apple store Genius Bar when possible for an evaluation.