Persistence beginning to rankle in annoying character (4)
I believe the answer is:
grit
'persistence' is the definition.
(thesaurus)
'beginning to rankle in annoying character' is the wordplay.
'beginning to' says to take the initial letters.
'in' means one lot of letters goes inside another.
'annoying character' becomes 'git' (informal term for an annoying person).
The initial letter of 'rankle' is 'r'.
'r' placed into 'git' is 'GRIT'.
(Other definitions for grit that I've seen before include "Toughness" , "Coarse stone or sand" , "Grains of rock or sand" , "Particles of stone/sand" , "Gumption or small pieces of stone" .)