Head off from state close to Connecticut (5)
I believe the answer is:
avert
'head off' is the definition.
(both can mean to prevent)
'state close to connecticut' is the wordplay.
'state' becomes 'aver' (both can mean to declare).
'close to' suggests the final letters (the close/ending of).
The last letter of 'connecticut' is 't'.
'aver'+'t'='AVERT'
'from' is the link.
(Other definitions for avert that I've seen before include "Fend off" , "Turn away or aside" , "Prevent, ward off" , "use preventive measures" , "Ward off (bad event)" .)