Great to see NZ added to Cycle Travel. Have cycled there the last 4 UK winters pre covid!
All I can do now is route plan for when I return. I noticed in planning a North Island route between Mangakino and Taupo that your route attempts to cross the Waikato River at Tram Road. Tram Road is private and I believe barred to cyclists for safety reasons (logging trucks!) - you can check the newsfeed on the Taupo Cycling Club website (admittedly a 2016 posting but no evidence that the status has changed)
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I know this is over 6 months old but I saw it had no replies and thought I'd add my thoughts.
I had this issue once or twice when living and riding in NZ, mostly with routes planned using komoot. But since I believe komoot also uses open street map data, this solution may be relevant here too.
If you have some local information, it can be useful to go over to https://www.openstreetmap.org/ and contribute any details such as a road being private, which I imagine route planners such as cycle.travel might take account of.
From a quick look, is it this road here: https://www.openstreetmap.org/way/836352661#map=19/-38.42001/175.95908
You could contribute to that, and it should be taken into account following an update to the database.