She grabs discarded seconds for European (4)

I believe the answer is:

serb

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'for european' is the definition.
(person from Serbia)

I cannot understand how the remainder of the clue works.

(Other definitions for serb that I've seen before include "A European national" , "From Belgrade, perhaps" , "One from Belgrade" , "Native of the Balkans" , "Eg, native of Belgrade" .)

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

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