Leaders having an interesting time in a Caribbean republic (5)
I believe the answer is:
haiti
'a caribbean republic' is the definition.
(I've seen this in another clue)
'leaders having an interesting time in' is the wordplay.
'leaders' suggests taking the first letters.
The first letters of 'having an interesting time in' is 'HAITI'.
(Other definitions for haiti that I've seen before include "Western part of island discovered by Columbus in 1492" , "Caribbean isle" , "Western half of Hispaniola" , "Country occupying part of Hispaniola" , "West Indies republic" .)