But... "So do you want _a_ specific thing, or will any thing work?" sounds kind of correct as well to me. (Disclaimer: I am not native speaker, grain of salt and all.)

If only it were that simple. "I bought a car yesterday" -- Did I buy a specific car?

a specific thing, or is it enough for it to be the thing?

sorry, just joking, this is not meant to be a genuine counterargument^^

I just want one of the new triple-cheese sandwiches please. And the salad of the day as a side.