Promotion from flier - 1/3 off! (4)
I believe the answer is:
puff
'promotion' is the definition.
(I've seen this before)
I don't understand how the rest of the clue works.
Can you help me to learn more?
(Other definitions for puff that I've seen before include "Drag on fag; be out of breath" , "out of which pants come?" , "Light pastry; advert" , "Gasp" , "plug" .)