Crossword Genius

Game that’s really high with a sauce on it (9)

Ross

I believe the answer is:

badminton

I'm a little stuck... Click here to teach me more about this clue!

'game' is the definition.
(badminton is a kind of game)

'really high with a sauce on it' is the wordplay.
I cannot really understand how this works, but
'a' is present in the answer.
'on' could be 'to' and 'to' is present in the answer.
This explanation may well be incorrect...

'that's' acts as a link.

Can you help me to learn more?

(Other definitions for badminton that I've seen before include "Racket game with shuttlecock" , "Game with shuttlecocks" , "in which rackets are involved" , "A sport; Beaufort seat" , "Game played with rackets" .)

I've seen this clue in the Evening Standard.
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