Hello All,
We wish to understand what sort of Audio Output is currently supported by CM5.
If we are required to use Audio Codec IC then is ES8388 supported ?
Otherwise if we are to use Audio_OUT_L and Audio_OUT_R, then how do we interface it with a headphone jack ?

First:
Your Codec chip needs the correct interfaces. You need I2S for the audio data and I2C for setting up the chip itself (internal registers).

Second:
You need a suitable Linux driver for the codec chip and you need to write a device tree overlay that matches your specific hardware. It's all in the Linux kernel. You need to have good knowledge of kernel programming for this. Otherwise you are going to fail.

Shortcut:
Search for a suitable Pi soundcard, look what codec is used there and work out the connections from the codec chip to the 40 pin header. Get the Device Tree Overlay source and use this as a foundation of your work.

Recommendation: TI PCM5122 seems to be well supported and is used on many Pi soundcards.

Would it work if I use the Audio_Out_L and Audio_Out_R pins from CM5 GPIO and connect them to the headphone jack through a series resistor and a capacitor to ground, instead of using the Audio Codec IC.

Should work as well, but with less quality. A dedicated codec chip has better audio quality.

Look for the Raspberry Pi4 Reduced schematics for the Audio filtering circuit.
6 resistors, 5 capacitors, and an 74LVC2G17 buffer chip.

ES8388 does have a Linux driver, so it should work. You could be the first person to test it on a CM5...

AUDIO_OUT_[LR] is available (GPIO pins 12, 13), but quality is a little lower (not quite Hi-Fi). See this reference design.

That filter circuit is not really a headphone driver, though it will work with higher-impedance headphones. For a stronger signal into low-impedance headphones, you'll need an amplifier. It depends on your application.