The Police Gazette records > there were several offences recorded > for James Henry COSSGROVE (surname also shown as COSGROVE) gave his birth year as about 1888 and also about 1891/92.
That type of variation in birth years, often occurred - as of course did miss-spellings of names.  
 ---
You may already have his birth listing from the NZBDM site ?
If not, is shows as
1891 - COSSGROVE - James Henry   (actual birthdate was  13 August 1891 )
son of  Kate* and William   [
*Some other COSSGRVE births - siblings ??  Mother shown as 
Katie - Father as William Henry  ?? ]
And this snippet I came across also (because he doesn't appear to have served in NZ Army WW1 ?) :
NZ Army - WW1  Reserve Roll - 2nd Division  :
COSSGROVE - James Henry -- Fisherman - Half Moon Bay, Stewart Island 
    ~  Lu