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I tried looking up a similar example to this sentence, but I couldn't:

図鑑読んで出直してきな

this isn't the whole sentence, the first part ends with たら conditional.

Eddie Kal
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legs
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    "this isn't the whole sentence" - Always give the whole sentence and more whenever possible. Wasn't a big deal here, but many times context is everything – By137 Aug 21 '18 at 15:35

1 Answers1

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You are parsing it incorrectly.

「出直{でなお}して・き・な」 = 「出直す」+「くる」+「な」

「な」 here is a sentence-ending particle that functions as an informal imperative-softener.

Thus, 「きな」 means virtually the same thing as 「来{き}なさい」

"Come again after reading the picture book, will you?"