Lively cheers following popular chief (2,4,7)
I believe the answer is:
in high spirits
'lively cheers following' is the definition.
Both the definition and answer are adjectives. Maybe there's an association between them I don't understand?
'popular chief' is the wordplay.
I cannot quite see how this works, but
'popular' could be 'in' ('in' can mean trendy) and 'in' is present in the answer.
The remaining letters 'highspirits' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...
Can you help me to learn more?
(Another definition for in high spirits that I've seen is " Elated".)