A child with a bad stye and coughing (6)

I believe the answer is:

chesty

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'and coughing' is the definition.
I can't tell whether this defines the answer.

'a child with a bad stye' is the wordplay.
'a child' becomes 'ch' (abbreviation for child).
'with' means one lot of letters go next to another.
'a bad' indicates anagramming the letters.
'stye' anagrammed gives 'esty'.
'ch'+'esty'='CHESTY'

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(Other definitions for chesty that I've seen before include "with a big top" , "a cougher?)" , "Having congested lungs" , "Scythe (anag)" , "well busted?" .)

I've seen this clue in the Evening Standard.

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