One oversensitive bit of skin on end of chin - it hurts shaving (9)
I believe the answer is:
snowflake
'skin on end of chin it hurts shaving' is the definition.
I can't judge whether this defines the answer.
'one oversensitive bit' is the wordplay.
'one oversensitive' becomes 'snow' (I am not sure about this - if you are sure you should believe this answer much more).
'bit' becomes 'flake' ('flake' can be a synonym of 'bit').
'snow'+'flake'='SNOWFLAKE'
'of' is the link.
Can you help me to learn more?
(Other definitions for snowflake that I've seen before include "Soft precipitation" , "Feathery construction of tiny ice crystals" .)