Persistent American set on seceding in the end (3-4)
I believe the answer is:
age-long
'persistent' is the definition.
(thesaurus)
'american set on seceding in the end' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'set' becomes 'gel' (both can mean to harden).
'in the end' suggests the final letters.
The last letter of 'seceding' is 'g'.
'a'+'gel'+'on'+'g'='AGE-LONG'
(Other definitions for age-long that I've seen before include "Having existed for many a year" , "Seeming to have always been so" .)