Crossword Genius

Look after bowling greens at the centre (7)

Ross

I believe the answer is:

oversee

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'look after' is the definition.
(both mean to have responsibility for)

'bowling greens at the centre' is the wordplay.
'bowling greens' becomes 'over' (I can't justify this - if you can you should give a lot more credence to this answer).
'at' means one lot of letters go next to another.
'the centre' becomes 'see' (see is a kind of centre).
'over'+'see'='OVERSEE'

Can you help me to learn more?

(Other definitions for oversee that I've seen before include "Inspect" , "guide" , "take charge" , "risk eyestrain?" , "survey" .)

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