Daughter dismissed from post by small English faculty (5)
I believe the answer is:
sense
'faculty' is the definition.
(sense is a kind of faculty)
'daughter dismissed from post by small english' is the wordplay.
'daughter' becomes 'd'.
'dismissed from' is a deletion indicator.
'post' becomes 'send' ('send' can be a synonym of 'post').
'by small english' becomes 'se' (I am not sure about this - if you are sure you should believe this answer much more).
'send' with 'd' taken out is 'sen'.
'sen'+'se'='SENSE'
(Other definitions for sense that I've seen before include "Feel; discernment" , "Feel intuitively" , "experience" , "Divine" , "Feel, perceive" .)