Conservative newspaper makes rough projection (4)
I believe the answer is:
crag
'rough projection' is the definition.
(I've seen this in another clue)
'conservative newspaper' is the wordplay.
'conservative' becomes 'C' (UK political abbreviation).
'newspaper' becomes 'rag' (informal term for a newspaper).
'c'+'rag'='CRAG'
'makes' is the link.
(Other definitions for crag that I've seen before include "Steep, rugged peak" , "Steep rock, peak" , "Steep rocky cliff" , "Steep, overhanging rock" , "Avant-garde" .)