Look after bowling greens at the centre (7)
I believe the answer is:
oversee
'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" .)