Stay and relax after a couple of problems initially (7)
I believe the answer is:
appease
'stay' is the definition.
('appease' can be a synonym of 'stay')
'relax after a couple of problems initially' is the wordplay.
'relax' becomes 'ease' (I've seen this before**).
'after' is a charade indicator (letters next to each other).
'couple of' indicates the doubling of some letters.
'initially' suggests taking the first letters.
The first letter of 'problems' is 'p'.
'p' duplicated is 'pp'.
'a'+'pp'='app'
'ease' put after 'app' is 'APPEASE'.
'and' is the link.
(Other definitions for appease that I've seen before include "Pacify with concessions" , "To behave like Chamberlain" , "Mollify, placate" , "be satisfying" , "Make calm by offering concessions" .)