Daughter and I very edgy, initially, in disreputable bar (4)
I believe the answer is:
dive
'disreputable bar' is the definition.
'daughter and i very edgy initially' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'and' says to put letters next to each other.
'initially' indicates taking the first letters.
The initial letters of 'very edgy' is 've'.
'd'+'i'+'ve'='DIVE'
'in' is the link.
(Other definitions for dive that I've seen before include "Jump into pool" , "Joint" , "Seedy bar" , "Plunge headfirst" , "Disreputable establishment" .)