Have to admit it hurt so close to recognition (3)
I believe the answer is:
own
'have' is the definition.
(synonyms)
'so close to recognition' is the wordplay.
'close to' suggests the final letters (the close/ending of).
'recognition' becomes 'wn' (I can't explain this - if you can you should believe this answer much more).
The final letter of 'so' is 'o'.
'o'+'wn'='OWN'
'to admit it hurt' acts as a link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for own that I've seen before include "Personal; possess" , "Possess - recognise - admit" , "Control" , "Private" , "Allow" .)