The power supply voltage does not affect the gain of an LM386 until the voltage is so low that the IC doesn't work properly.
The datasheet shows the output at clipping is 0.45W into 8 ohms with a 9V supply. With a 12V supply the output is slightly higher but the heat is much higher.
The power supply voltage also does not affect THD unless it is driven into clipping.
The voltage from a little 9V battery drops as it is used so of course the amount of output power at clipping also reduces.
The amplifier has a very high internal gain and has negative feedback that reduces the gain to 20 or 200. The high amount of negative feedback results in low harmonic distortion.