Your JST plug cables have four wires: black, red, white, yellow. The colours on these cables do not mean power and ground the usual way. What each wire carries depends on which socket it's plugged into. Everything below is read from the labels printed on your board.
The one rule: on the CN1 cable, red is a signal (IO22), not power, and yellow is 3.3 V. On the P3 cable, yellow is IO21, the screen backlight, so never connect it. Go by this page, not by the wire colour.
Your board, labelled
The cable plugs in only one way, so each colour always lands on the same pin. Black sits on the GND pin on all three sockets.
What each wire does
P1 · power in
yellowVIN ← power board 5V+
blackGND ← power board 5V−
redRX · don't cut: insulate the end
whiteTX · don't cut: insulate the end
CN1 · mic + amp
blackGND → mic GND
redIO22 → mic SCK + amp BCLK
whiteIO27 → mic WS + amp LRC
yellow3.3V → mic VDD
P3 · mic data
redGPIO35 → mic SD (printed IO33, a misprint)
blackGND · cut short, tape
whiteIO22 · cut short, tape
yellowIO21 · backlight, never use
Diagram 1 · Power
The battery feeds the power board. Its 5V output goes to two places: the screen through the P1 cable, and the amp. The cable wires are short, so splice your red and black 22 AWG wire onto them (● = solder joint with heat-shrink).
Diagram 2 · Audio (mic and amp)
CN1 and P3 feed the mic. IO22 and IO27 also go to the amp, so those two wires split. The IO4 pad wire (plain white) goes straight to the amp's DIN pin.
Every connection, in one list
From
Cable wire
Your 22 AWG
To
Power board 5V+
P1 yellow
red
Screen VIN (via P1 cable)
Power board 5V−
P1 black
black
Screen GND (via P1 cable)
Power board 5V+ / 5V−
none
red / black
Amp VIN / GND
CN1 GND
black
black
Mic GND (+ L/R jumper)
CN1 IO22
red
blue, splits
Mic SCK + Amp BCLK
CN1 IO27
white
green, splits
Mic WS + Amp LRC
CN1 3.3V
yellow
orange
Mic VDD
P3 2nd contact (GPIO35)
red
yellow
Mic SD
IO4 resistor pad
none
white
Amp DIN
Battery + / −
none
holder wires
Power board B+ / B−
Button
none
yellow + tape / black
Power board K / B−
Speaker + / −
none
red / black
Amp green terminal + / −
Unused: insulate each end
P1 red + white (keep full length, insulate: handy for serial debugging) · P3 black + white + yellow (IO21)
Check before you power it
P1 colours. Plug a cable into P1 and one into CN1, with no battery connected. Multimeter on beep: touch the black wire of the P1 cable and the black wire of the CN1 cable. It should beep, because both are GND. If it doesn't, the P1 cable is the other way round, so stop and send me a photo.
5V at the cable. With the P1 cable unplugged from the screen, put the cells in, press the button, and measure between the cable's yellow and black: about 5.0–5.2 V, yellow positive. Then cells out and plug it in.
3.3V at CN1. With everything plugged in and powered, measure CN1 yellow to black: about 3.3 V. That's the mic's power. Never put 5V on this wire.
Splices. Slide heat-shrink on before soldering, and shrink it over every joint. The cut, unused wires get a wrap of tape each so they can't touch anything.