I have a mobile phone (SE K750i) with a 3.5mm audio adapter to connect
to a 3.5mm audio output source, which enables me to redirect the
volume from the phone to headphones or external speakers. However,
when I do this, I lose the microphone function of my phone. Is there a
way I can split the audio output to connect an external microphone to
the phone?

While I accept that an audio out will normally not support a
microphone, my phone came with a standard headset consisting of
earphones and a built-in microphone, which plugs into the same socket
where the new audio adapter goes.