Rushed to the right-hand side (4)
I believe the answer is:
tore
'rushed' is the definition.
(to tear around is to rush or run fast)
'the right-hand side' is the wordplay.
'the' becomes 't' (the is pronounced as a 't' sound in some dialects).
'right-hand side' becomes 'ore' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
't'+'ore'='TORE'
'to' is the link.
Can you help me to learn more?
(Other definitions for tore that I've seen before include "came apart" , "Pulled something apart forcefully" , "Ripped, ruptured" , "'Ripped, rent (4)'" , "Rushed" .)