I threw fan out for being so skinny (5,4)

I believe the answer is:

wafer thin

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'being so skinny' is the definition.
I know nothing about this answer so I cannot tell whether this works.

'i threw fan out' is the wordplay.
'out' is an anagram indicator (out can mean wrong or inaccurate).
'i'+'threw'+'fan'='ithrewfan'
'ithrewfan' is an anagram of 'WAFER-THIN'.

'for' acts as a link.

Can you help me to learn more?

(Other definitions for wafer thin that I've seen before include "Narrow" , "Having very little width or substance" , "Much pinched" , "Unconvincing" , "Like something on ice" .)

I've seen this clue in The Guardian.

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