Firm importing trimmed version of fat cigar (5)

I believe the answer is:

claro

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'fat cigar' is the definition.
'claro' can be an answer for 'cigar' (claro is a kind of cigar). I'm not certain of the 'fat' bit.

'firm importing trimmed version' is the wordplay.
'firm' becomes 'co' (a firm is a company).
'importing' is an insertion indicator.
'trimmed version' becomes 'lar' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'co' enclosing 'lar' is 'CLARO'.

'of' is the link.

Can you help me to learn more?

(Another definition for claro that I've seen is " cigar".)

I've seen this clue in The Independent.

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