Person in authority that must be straight! (5)
I believe the answer is:
ruler
This clue is a double definition.
'person in authority' is the first definition.
(someone who rules or governs)
'that must be straight' is the second definition.
(a ruler for measuring things must be straight)
(Other definitions for ruler that I've seen before include "Measuring device" , "liner?" , "Drawing stick or monarch" , "Controller" , "Chief" .)