Crossword Genius

I'm leaving food that's picked up (4)

Ross

I believe the answer is:

ciao

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

'that's picked up' is the definition.
I can't judge whether this definition defines the answer.

'i'm leaving food' is the wordplay.
I cannot really understand how this works, but
'i'm' could be 'i' (I've seen this in other clues) and 'i' is found in the answer.
'food' could be 'a' (A is an example) and 'a' is found in the answer.
The remaining letters 'co' is a valid word which might be clued in a way I don't understand.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?

(Other definitions for ciao that I've seen before include "parting word" , "Italian hello-goodbye" , "Hello and goodbye to the Italians" , "European's greeting" , "Italian's so long" .)

I've seen this clue in The Guardian.
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