Hi Harley08
Welcome to RootsChat.

Am presuming that you already have the death certificate for Clara ... and that it doesn't give you the names of her parents either ?
It is of course unlikely that she was born in New Zealand (the earliest recorded "CHARLTON" birth is in 1859).
From a quick search of passenger lists into New Zealand, I'm afraid that I couldn't see anything for a CHARLTON family nor for Clara, arriving prior to 1864.
Regarding whether Clara, James Edward and son John Leonard, ever "came out to Australia", you would need to post a query on the Australia board, for this.
John Leonard appears to have been born in NZ in 1881 and there are 2 further children with parents James Edward and Clara listed on the online birth index.
1880 - Maria Louise MILLS ... and ... 1883 - Frederick Powley MILLS. (There a several more births with parents named as just "James and Clara" who may also be connected ?)
You can view the index here >>>
http://bdmhistoricalrecords.identityservices.govt.nz/Lu