Stick around the centre of Geneva (4)
I believe the answer is:
cane
'stick' is the definition.
(cane is a kind of stick)
'around the centre of geneva' is the wordplay.
'around' becomes 'ca' (circa).
'the centre of' means to look at the middle letters.
The centre of 'geneva' is 'ne'.
'ca'+'ne'='CANE'
(Other definitions for cane that I've seen before include "Staff < with walking-stick" , "Type of sugar; beat" , "Slender support, for 32 across say" , "not beet" , "Slender support for a plant" .)