OK, it was just the angle of the first photo, with the long leads obscuring things.
I'd expect an external mic to connect to the same preamp as the internal one, switched by the socket contacts. Trying an external mic would be helpful is possible?
The transistor circuit near the mic could be the preamp?
(I initially though the big I could include audio processing - but seeing the number, it's the LED display driver).
That white component is probably a ceramic capacitor for RF noise suppression. You should be able to use solder wick to clean up the joints and remove the mic. When refitting it, use a toothpick to hold it in place while you reflow one end.
First, though - the big surface mount module with the screening can looks to have some dubious joints. It may be worth going along those and adding a bit more solder. If one is the analog audio, a cracked joint could easily cause problems.