Weary former lover, American, in Homburg? (7)
I believe the answer is:
exhaust
'weary' is the definition.
(exhausting is a kind of wearying)
'former lover american in homburg?' is the wordplay.
'former lover' becomes 'ex' (ex-lover).
'american' becomes 'US' (abbreviation for United States).
'in' means one lot of letters goes inside another.
'homburg?' becomes 'hat' (homburg is a kind of hat).
'us' put within 'hat' is 'haust'.
'ex'+'haust'='EXHAUST'
(Other definitions for exhaust that I've seen before include "part of car - > tire?" , "Car's fume pipe" , "Devitalize, drain of energy" , "Pipe to take away car fumes" , "Tube carrying off fumes from car engine" .)