Reduction in what roughly batmen eat (9)

I believe the answer is:

abatement

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'reduction' is the definition.
(abatement is a kind of reduction)

'roughly batmen eat' is the wordplay.
'roughly' is an anagram indicator.
'batmen'+'eat'='batmeneat'
'batmeneat' is an anagram of 'ABATEMENT'.

'in what' acts as a link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.

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(Other definitions for abatement that I've seen before include "Subsidence" , "relief" , "Lessening in intensity or amount" , "mitigation" .)

I've seen this clue in the Sydney Morning Herald.

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