Self-absorbed, say, with wild conceit about source of reputation (10)
I believe the answer is:
egocentric
'self-absorbed' is the definition.
(I've seen this before)
'say with wild conceit about source of reputation' is the wordplay.
'say' becomes 'eg' (**).
'with' means one lot of letters go next to another.
'wild' indicates an anagram.
'about' says to put letters next to each other.
'source of' indicates taking the first letters.
The initial letter of 'reputation' is 'r'.
'conceit' put after 'r' is 'rconceit'.
'rconceit' with letters rearranged gives 'ocentric'.
'eg'+'ocentric'='EGOCENTRIC'
(Other definitions for egocentric that I've seen before include "Self-seeking" , "Self-interested" , "Self-centred person with little regard for others" , "Self-absorbed" , "Self-obsessed" .)