Sound spectrum analysis, adverse harmonic tones, micropython, adafruit, fft
As you noted, and the third plot accentuates this, the spacing is always very close to 1 kHz, but just offset by something like 240 Hz.  This is indicative of an alias signal.  It would be against a harmonic of the sample rate (guessing to be 10 kHz).  I'd get a broadband scope if possible and put a 
loop of wire on it ( a circle a couple of inches in diameter) to "sniff" for signals.  The signal might be from the board, or it might be from your monitor, the PC power supply, an LED or fluorescent bulb or anything with power.