David's father's making straps for falconers (6)

I believe the answer is:

jesses

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'falconers' is the definition.
I know nothing about this answer so I can't judge whether it can be defined by this definition.

'david's father's making straps' is the wordplay.
'david's father' becomes 'jesse' (I've seen this in another clue).
'making straps' becomes 's' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'jesse'+'s'='JESSES'

'for' acts as a link.

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I've seen this clue in The Times.

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