Creating a custom TRRS splitter for headphones and mic; how to handle the mic's TRS jack?
The de facto standard for PC microphone power and audio is commonly called “plug-in power”. You can find some documentation of it using that search term.

Sleeve: common
Ring: 3-5 V power, current limited through a series resistor
Tip: audio, AC coupled through a series capacitor

In practice, since electret microphone capsules have only two contacts and expect the power combined with the audio, tip and ring are often shorted together by the wiring of the microphone.
Some sound cards support stereo microphones by using tip and ring as two channels each with combined power and audio; this is backwards compatible since the microphone will get power from one channel and send audio to the other or both.

In your case, since you have only the two wires of a TRRS headset connection's microphone circuit, you have two options:

Wire the TRRS sleeve microphone channel to the TRS jack's tip and ring. This is suitable for electret microphones.
Insert a DC-blocking capacitor in series with the tip connection and a resistor in series with the ring connection. This will enable compatibility with un-powered microphones or other audio sources using a TS connection, whereas the first option will be shorted out by a TS plug (since TS is equivalent to TRS with ring shorted to sleeve). However, it might impair the audio quality or level since you are adding additional impedances on both lines.