Hi Nickyhug
You will have seen on the Auckland Cenotaph website, that Edwin Simpson ANDREWS is listed as having served (WW1) in the Pacific region, in the Samoa Campaign. That he is in the Brockenhurst area in 1918, tends to suggest that he also saw service, further abroad.
His war service record would include details of hospitalisation (usually just a brief description of illnesses, wounds suffered etc.). WW1 records are held by Archives New Zealand at their Wellington office and the great majority, are listed with a file number reference, on the Archives NZ "Archway" website. Unfortunately, there doesn't appear to be a record for Edwin Simpson ANDREWS.
[You may though, like to send a query to Archives NZ to check further].
Obviously finding the children of this couple, is not easy. (You appear a little uncertain on your other thread, as to whether the couple actually returned to New Zealand)

Also, with a surname like ANDREWS, we would require further information such as second christian names (if any), for the children.
Lu