Street kid should wait if it's lost (4)

I believe the answer is:

waif

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'street kid should' is the definition.
I can't judge whether this defines the answer.

'wait if it's lost' is the wordplay.
'lost' suggests deleting specific letters.
'wait'+'if'='waitif'
'waitif' with 'it' taken away is 'WAIF'.

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(Other definitions for waif that I've seen before include "Gamin" , "An unowned or abandoned child" , "Homeless child - thin person" , "Forsaken child" , "Ragamuffin" .)

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

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