Woman with aim to get around India (4)
I believe the answer is:
enid
'woman' is the definition.
(woman's name)
'aim to get around india' is the wordplay.
'aim' becomes 'end' (both can mean a goal or purpose).
'to get around' indicates putting letters inside.
'india' becomes 'i' (phonetic alphabet: alpha, bravo, charlie etc.).
'end' going around 'i' is 'ENID'.
'with' acts as a link.
(Other definitions for enid that I've seen before include "Geraint's loved one" , "'... Blyton, children's author (4)'" , "Dine out on this girl's name also" , "Miss Blyton" , "- Blyton, twentith century writer" .)