Those people after working with purpose must be improving (2,3,4)
I believe the answer is:
on the mend
'improving' is the definition.
(I've seen this before)
'those people after working with purpose' is the wordplay.
'those people' becomes 'them' (I've seen this before).
'after' means one lot of letters go next to another.
'working' becomes 'on'.
'with' is a charade indicator (letters next to each other).
'purpose' becomes 'end' (I've seen this before).
'them' put after 'on' is 'onthem'.
'onthem'+'end'='ON THE MEND'
'must be' is the link.
(Other definitions for on the mend that I've seen before include "convalescing" , "Informally getting better, improving after illness or accident" , "Improving in health" , "undergoing recovery?" , "Becoming better in health" .)