Rook heading to the king's file gains a tempo (4)
I believe the answer is:
rate
'tempo' is the definition.
(I've seen this in another clue)
'rook heading to the king's file gains a' is the wordplay.
'heading to' suggests taking the first letters.
'the king's file' becomes 'te' (I can't justify this - if you can you should believe this answer much more).
'gains' means one lot of letters goes inside another.
The initial letter of 'rook' is 'r'.
'r'+'te'='rte'
'rte' going around 'a' is 'RATE'.
Can you help me to learn more?
(Other definitions for rate that I've seen before include "standard levy" , "Assess; speed" , "Amount measures against another" , "Regard" , "The number of times something happens in a given period" .)