I just had this same looking and sounding problem.https://forum.bambulab.com/t/what-is-this-clicking-by-my-extruder/89238
Check the screws on the Hotend Heating Assembly. My problem was fixed by the second part of this wiki
(https://wiki.bambulab.com/en/a1-mini/troubleshooting/print-issues-troubleshooting)
There are four screws on the back of the Heating Assembly that hold its 2 parts together. When I opened mine up, one screw pop-out at me (completely undone) and the other 3 were loose too. The wiki has pictures and instructions for tightening the screws.
Might be worth a check. I had never taken the Hotend Heating Assembly off my machine before so it was like this from the day I got it.