if a screw belongs in a hole it will be threaded .. and at the bottom of the hole will be an orifice of some kind that the tip of the needle-screw fits into.
maybe you're looking at some hole other than a mixture screw hole.. Carbs are drilled and plugged in several places for various reasons, although its very rare for a plug to come out on it's own..
got a pic? or know a web page with a good view of the carb?
Part 2 of the QT manual (Yamaha QT50 Shop Manual Part 2(PDF)) on THIS page ..
Page 4-1 (page 23 of 30 in the pdf file) has a good exploded view of the carb.. is that the carb?
I see nothing that coulda broken off the carb bodyt down at the "pilot air screw" hole..
Maybe the original screw is actually in there and it was turned in too deep into the hole? Or the screew broke in two?