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Hi,
could you tell which sentence is correct?
1) "She gave extra cake away to ........"
2) "She gave an extra cake away to ........"

Thank you
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Magda, you're referring to specific cake - the cake that was given away.

If she made an entire extra cake to give away, you can say "She [made two cakes for us to have at the party and] gave an extra cake to the family that wasn't able to come."

OR

There was leftover (extra) cake from the event, in which case, she gave away the extra cake, using in the uncountable sense.

We had a birthday party for Jeff at work. Nancy arranged to buy this huge cake from the store. Even after we all had second pieces, a lot was left over. So we gave the extra cake to the sales department.
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Comments  
First I wrote the sentence without an article, than I started to think that maybe "an" is needed. After all, "cake" is countable. On the other hand, I found on the internet many sentences where, in a similar context, there was no article. So, I am a bit confused.
Both correct, depending on the meaning.

Made in large shapes, to be cut in portions: uncountable
Made in small pieces, one per person: countable

It's both C (Countable) and U (uncountable). This is the dictionary you should be using for such issues:

http://dictionary.cambridge.org/define.asp?key=10780&dict=CALD
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 BarbaraPA's reply was promoted to an answer.
Thank you a lot Emotion: smile