Will turtle beach give a full refund in cash for a headset I've used once?

I also think the acu or mix amp may be broken

No

The store you bought if from would issue the refund not Turtle Beach. If there is a problem with it take it to the store you brought if from. They have to fix or replace it under the manufacturer one year warranty. If you bought it on line you would have to contact Turtle Beach and ask how the warranty works.