A very tricky question indeed.
Let me translate your sentences to English so you can observe the difference.
- ¿Qué has perdido? / What have you lost?
- ¿Que has perdido qué? / You have lost what!?
Notice that in the Spanish version, only the second qué has the accent mark indicating that it is a pronoun instead of a conjuction.
To ask about what has been lost, you have to use the first sentence
¿Qué has perdido?
If you have not hear what has been lost or you can't believe that such thing can be lost so easily and want to show your amazement or angriness, use the second option
¿Que has perdido qué?
Dialogue example 1
- I've lost my ... (inaudible)
- What? You have lost what?
- My phone.
In Spanish it would be
- He perdido mi ... (inaudible)
- ¿Que has perdido qué?
- Mi teléfono.
Dialogue example 2 (angriness)
- I've lost your phone. Sorry.
- You have lost what!?? How??
In Spanish it would be
- He perdido tu teléfono. Lo siento.
- ¿Que has perdido qué!?? ¿Cómo??