Process hides most of President's petulant display (7)
I believe the answer is:
tantrum
'petulant display' is the definition.
'tantrum' can be an answer for 'display' (I have seen 'Display of temper' mean 'tantrum' so perhaps 'display' could also mean 'tantrum'). I am unsure of the 'petulant' bit.
'process hides most of president's' is the wordplay.
'process hides' becomes 'tan' (I can't justify this - if you can you should give a lot more credence to this answer).
'most of' means to remove the last letter.
'president' becomes 'trump' (Donald Trump).
'trump' with its last letter taken away is 'trum'.
'tan'+'trum'='TANTRUM'
Can you help me to learn more?
(Other definitions for tantrum that I've seen before include "Burst of temper" , "Fit of childish temper" , "Childish angry outburst" , "Uncontrolled outburst of anger" , "Childish fit of rage" .)