Crossword Genius

Something cylindrical players will fight for? (7)

Ross

I believe the answer is:

bandbox

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

'something cylindrical' is the definition.
(circular carrier for hats)

'players will fight for?' is the wordplay.
I cannot really understand how this works, but
'players' could be 'band' (I've seen this before) and 'band' is found in the answer.
'fight' could be 'box' (boxing is a kind of fighting) and 'box' is present in the answer.
No letters remain.
This explanation may well be incorrect...

Can you help me to learn more?

(Another definition for bandbox that I've seen is " Carrier for hats".)

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