Strange fatigue besetting millions — what smoking may do (8)
I believe the answer is:
fumigate
'what smoking may do' is the definition.
(as in fumigation)
'strange fatigue besetting millions' is the wordplay.
'strange' indicates anagramming the letters.
'besetting' means one lot of letters goes inside another.
'millions' becomes 'm' (abbreviation).
'fatigue' with letters rearranged gives 'fuigate'.
'fuigate' placed around 'm' is 'FUMIGATE'.
(Other definitions for fumigate that I've seen before include "Treat (pests) with vapour" , "Disinfect using smoke or gas" , "Kill with, eg, smoke" .)