Cigarette lit, fire heartlessly put out (6,3)
I believe the answer is:
filter tip
'cigarette' is the definition.
(I've seen this before)
'lit fire heartlessly put out' is the wordplay.
'heartlessly' suggests removing the centre.
'out' is an anagram indicator (out can mean wrong or inaccurate).
'put' with its centre taken out is 'pt'.
'lit'+'fire'+'pt'='litfirept'
'litfirept' anagrammed gives 'FILTER TIP'.
(Other definitions for filter tip that I've seen before include "A fag" , "Tobacco purifier" , "Cigarette feature" , "Cigarette with an attachment for removing some of the harmful substances" .)