Separate people struggle for universal supremacy (5,4)
I believe the answer is:
space race
'struggle for universal supremacy' is the definition.
I know nothing about this answer so I can't judge whether it can be defined by this definition.
'separate people' is the wordplay.
'separate' becomes 'space' (I can't justify this - if you can you should believe this answer much more).
'people' becomes 'race' (people can mean a national group or race).
'space'+'race'='SPACE RACE'
Can you help me to learn more?
(Another definition for space race that I've seen is " Aerial competition".)