Daughter, extremely embarrassed, welcomes facial hair being removed (8)
I believe the answer is:
detached
'being removed' is the definition.
(thesaurus)
'daughter extremely embarrassed welcomes facial hair' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'welcomes' is an insertion indicator (in sense of 'invites in').
'facial hair' becomes 'tache' (I've seen this before).
'embarrassed' with its middle removed is 'ed'.
'ed' enclosing 'tache' is 'etached'.
'd'+'etached'='DETACHED'
(Other definitions for detached that I've seen before include "Not committed, objective" , "Separated from - maybe a house" , "Separate; aloof" , "Clinical" , "Separate, disconnected" .)