Bird grabs the lady's cigar (7)
I believe the answer is:
cheroot
'cigar' is the definition.
(cheroot is a kind of cigar)
'bird grabs the lady's' is the wordplay.
'bird' becomes 'coot' (coot is a kind of bird).
'grabs' is an insertion indicator.
'the lady' becomes 'her' (her refers to a woman).
'coot' placed around 'her' is 'CHEROOT'.
(Other definitions for cheroot that I've seen before include "Havana?" , "With both ends cut off squarely, is smoked" , "'Cigar, square at both ends (7)'" , "Variety of cigar" , "Small cigar" .)