Thanks to you all for suggestions made which, I believe, has now enabled me to crack it.
The Auckland Cenotaph Database states "Troopships Arawa or Limmerick."
But a clearer copy of Samuel's Service Record than anything I had yet seen, showed Tsport. 3
leaving NZ 15/8/14 disembarking in Alexandria on 14/10/14 after 50 days
Arawa was No.10, Limerick was No.7 and No. 3 was Maunganui.
The book "Wellington Regiment (NZEF) 1914-19". by Cunningham et al is very helpful.
If there is an error in my reasoning, I shall be pleased to hear. Many thanks, T.