Nightingale, famously put game back within three points (5)
I believe the answer is:
nurse
'nightingale famously' is the definition.
(Florence Nightingale is an example)
'game back within three points' is the wordplay.
'game' becomes 'ru' (rugby union).
'back' shows that the letters should be reversed in order.
'within' indicates putting letters inside.
'three points' becomes 'nse' (**).
'ru' written backwards gives 'ur'.
'ur' put within 'nse' is 'NURSE'.
'put' is the link.
I am not very happy about this link. It may belong to another bit of the clue.
(Other definitions for nurse that I've seen before include "Hospital worker" , "Tend to patient" , "Suckle" , "Sister, e.g." , "One ministering to the sick" .)