The sweet potato made it from South America to the New Zealand at or just after the time the first Māori arrived in the Land of the Long White Cloud. Obviously we don't know exactly where and when the Polynesians visited South America to collect sweet potato, but we know that they did, somehow. The potato could potentially come across at the same time.
Okay, from what little I've seen, the introduction of the kumara into Polynesia was before the colonisation of Aotearoa, especially since its colonisation seems to be about 1300 AD, while there have been kumara remains in Hawai'i dating from 1100 AD. I wouldn't be surprised if the Polynesians did try to introduce the potato. However, since the tropics generally aren't very good for potato growth, I also wouldn't be surprised if they gave up on it.
Whether or not the Polynesians tried to introduce the potato as well as the kumara, this would then mean that we'd need another trip to the Americas to (re)introduce the potato. To achieve this, we're gonna need a motivation for long distance travel.
Fortunately, there is such a motivation: in about the 1500's a group of Polynesians, possibly Maori, maybe not, migrated from the South Island to Rekohu/Chatham Islands and became the Moriori. Perhaps from there, they could do a quick multi-month trip to South America. I'm not really sure how that would lead to the introduction of the potato into the South Island, but this probably would lead to an increase in Maori population.
