Very inaccurate story about religious coverage (4)
I believe the answer is:
veil
'religious coverage' is the definition.
'very inaccurate story about' is the wordplay.
'very' becomes 'v' (abbreviation).
'inaccurate story' becomes 'lie'.
'about' says the letters should be written backwards.
'lie' written backwards gives 'eil'.
'v'+'eil'='VEIL'
(Other definitions for veil that I've seen before include "Nun's head-covering" , "Material over face" , "Covering for head and face" , "Yashmak, e.g." , "Face concealer" .)