Carelessly mention a name (8)
I believe the answer is:
nominate
'name' is the definition.
(I know that name can be written as nominate)
'carelessly mention a' is the wordplay.
'carelessly' is an anagram indicator.
'mention'+'a'='mentiona'
'mentiona' is an anagram of 'NOMINATE'.
(Other definitions for nominate that I've seen before include "Put forward (as candidate for election)" , "Designate" , "Pick as a candidate" , "Choose" , "Specify (person for job)" .)