robinm ok so - i really couldn't hear anything from the first circuit (no baseline?), but the second circuit (the one including the MEMS) actually DOES work like a charm
That's a good first step.
robinm The microphone itself is still crunchy as crazy, but the baseline noise has been practically completely removed this way.
Is this the case both when powering from USB or from LiPo?
To assess and address the distortion you are hearing, can you show the microphone signal on the Bela scope? In Pd you 'd do:
[adc~ 3] << mic input: analog in 0
|\\
| | \
| | \
| [dac~ 1 2] << output to headphones. Does it distort when powered via USB?
|
[dac~ 27] << send to scope
Then view it at http://bela.local/scope . Show screenshots both with the microphone connected and without it (without C2).
It is harder/impossible to use the Scope when powered via LiPo. If the issue only presents itself only with the LiPo, consider using a program that can record the analog input as a wav file to disk and send that wav file over. In Pd it would be:

Set this program to run on boot, power via LiPo, let it run for ten seconds, then power it off gracefully, power via USB, open up the IDE, download the wav file and look at it / post it here.
Since it seems too much like magic to believe it, can you explain me what's happening?
I don't know exactly how you were wiring the microphone before. This circuit provides a bias of about 2V at the analog input which works as the central point, which is what the two resistors do. C1 removes some of the high frequency noise. When you add the microphone, C2 ensure the signal is AC-coupled and no DC-voltage at the output of the microphone affects the 2V bias. If the mic breakout already has an output capacitor, this could be removed. The AC-coupled signal from the microphone will be added to the 2V DC bias and so it will oscillate around the 2V offset, thus remaining within the 0V : 4.096V range of the ADC.
Now, I just looked at the circuit for your MEMS breakout: https://cdn.sparkfun.com/assets/c/3/7/5/e/SparkFun_Analog_MEMS_Microphone_Breakout_ICS40180_Schematic.pdf . If this is accurate, then the output of the breakout should already be centered at around 1.65V and so the diagram I suggested shouldn't be needed, at least for basic operation (although it may improve the dynamic range when Bela is powered from USB). So if you have something that works with my circuit that didn't work before, it may be one of these options:
- you wired the previous circuit wrong (although it should be straightforward)
- you wired my circuit wrong, something is hitting the rail (GND or 5V) when no microphone is connected, thus saturating the ADC input (dead silence!) and causing the subsequent distortion when the microphone is indeed connected. Looking at the scope and/or recorded wav file may shine some light on this
- something else that requires further investigation