Crossword Genius

Commonly take the chill off and not leave a scrap (3,2)

Ross

I believe the answer is:

eat up

I'm a little stuck... Click here to teach me more about this clue!

'scrap' is the definition.
Although both the answer and definition are verbs in their base form, I cannot understand how they can define each other.

'commonly take the chill off and not leave a' is the wordplay.
I cannot really see how this works, but
'take' could be 'eat' (eating is a kind of taking) and 'eat' is found within the answer.
The remaining letters 'up' is a valid word which might be clued in a way I don't see.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?

(Other definitions for eat up that I've seen before include "Exhaust" , "polish off" , "finish meal" , "show good appetite" , "Consume completely" .)

I've seen this clue in The Guardian.
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