Crossword Genius

Possible battle with fussy kid over veg at duke’s grassland? (10)

Ross

I believe the answer is:

greensward

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'possible battle with fussy kid' is the definition.
I can't tell whether this definition defines the answer.

'veg at duke's grassland?' is the wordplay.
'veg' becomes 'greens' (greens is a kind of vegetable).
'at' says to put letters next to each other.
'duke's grassland?' becomes 'ward' (I can't justify this - if you can you should give a lot more credence to this answer).
'greens'+'ward'='GREENSWARD'

'over' is the link.

Can you help me to learn more?

(Other definitions for greensward that I've seen before include "mostly turf" , "Grass covered ground" , "Lawn" .)

I've seen this clue in The Times.
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