Hello & Welcome to TSF: :wave:
You could put your 20 pin connector into your 24 pin motherboard and it will fire up just align the snap lock and it will be ok
however a 350 watt power supply is VERY under sized to run a more updated motherboard spec (which yours must be or it wouldnt be a 24 pin)
please give us your full ssytem specs and we can make some recommendations for your