Angry about navy's nocturnal annoyance (5)
I believe the answer is:
snore
'nocturnal annoyance' is the definition.
(I've seen this in another clue)
'angry about navy's' is the wordplay.
'angry' becomes 'sore' (similar in meaning).
'about' is an insertion indicator.
'navy' becomes 'N' (common abbreviation eg in RN Royal Navy).
'sore' enclosing 'n' is 'SNORE'.
(Other definitions for snore that I've seen before include "Noise from a sleeper" , "Make raucous nasal sound during sleep" , "Noise made while asleep" , "Sound made by sleeper" , "thing which bores" .)