Want to get under British player trying to mate? (5)
I believe the answer is:
black
'player trying to mate?' is the definition.
(a chess player tries to checkmate their opponent)
'want to get under british' is the wordplay.
'want' becomes 'lack' (synonyms).
'to get under' means one lot of letters go next to another (in a down clue, some letters go under others).
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'lack' put after 'b' is 'BLACK'.
(Other definitions for black that I've seen before include "second player on board" , "Served without milk" , "Chess player" , "What's written on in the classroom" , "Boycott" .)