An American gives no backing to king (4)
I believe the answer is:
yank
'an american' is the definition.
(Yank can informally mean an American)
'no backing to king' is the wordplay.
'no' becomes 'nay' (archaic form of 'no').
'backing' says the letters should be written backwards.
'to king' becomes 'K' (abbreviation for king in chess).
'nay' written backwards gives 'yan'.
'yan'+'k'='YANK'
'gives' is the link.
(Other definitions for yank that I've seen before include "Sharp tug" , "American from the North" , "Jerk, pull suddenly" , "Tug sharply - American" , "Pull with a sudden movement" .)