Go after — and get — piece for a mate (4)
I believe the answer is:
seek
'go after' is the definition.
(thesaurus)
'piece for a mate' is the wordplay.
'piece' becomes 'k' (abbreviation for King).
'for' means one lot of letters go next to another (I've seen this in other clues).
'a mate' becomes 'see' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'k' after 'see' is 'SEEK'.
'and get' acts as a link.
Can you help me to learn more?
(Other definitions for seek that I've seen before include "Attempt - search" , "Attempt to find; ask for" , "Go looking for" , "Hunt, look for" , "Try to locate or reach" .)