Not entirely remarkable English story (4)
I believe the answer is:
tale
'story' is the definition.
(synonyms)
'not entirely remarkable english' is the wordplay.
'not entirely' means to remove the last letter (most but not all of the word).
'remarkable' becomes 'tall' (I've seen this before).
'english' becomes 'E' (abbreviation as in OED).
'tall' with its last letter removed is 'tal'.
'tal'+'e'='TALE'
(Other definitions for tale that I've seen before include "Story; lie" , "An account of incidents or events" , "Late with a story" , "relation" , "Old wives' recital?" .)