Prejudice shown by one in HQ out east (4)
I believe the answer is:
bias
'prejudice' is the definition.
('bias' can be a synonym of 'prejudice')
'one in hq out east' is the wordplay.
'one' becomes 'i' (Roman numeral**).
'in' means one lot of letters goes inside another.
'hq' becomes 'base' (I've seen this in other clues).
'out' indicates named letters should be taken away.
'east' becomes 'e' ('E' can be a synonym of 'east').
'base' with 'e' taken away is 'bas'.
'i' going within 'bas' is 'BIAS'.
'shown by' acts as a link.
(Other definitions for bias that I've seen before include "A leaning or prejudice" , "Favouritism" , "(Bowls) irregular shape" , "could be binding" , "An influence against, prejudice" .)