Clark was seventh equal of the nine remaining candidates, attracting six "encourage" votes and nine "discourage" votes.
That was an increase in discourage votes from the previous poll this month, when Clark had seven "discourage" and two "no opinion" votes.
In today's poll, for the first time, she got no "no opinion" votes.
Soon after today's poll, Clark described it as a "another positioning poll" and said she would continue her campaign.
"We did not expect the results to be very different from the previous poll and this turned out to be the case.
"There were more 'discourage' votes exercised against most candidates in this round. I am still in a group with most other candidates.
"I am pressing ahead towards the next poll."
The results indicate Clark faces vetoes from one or more of the five Permanent Members on the Security Council. They are China, France, Russia, the United Kingdom, and the United States.
That will become clear when the first poll in which the permanent five will use coloured ballots is taken early next month.
In today's results former Portugal Prime Minister Antonio Guterres remained in the top slot with 12 encourages and only one discourage.
The result will disappoint Clark and Key, who was in New York campaigning alongside Clark just last week to boost her chances.
Key said it was possible Clark would attract vetoes but that would not necessarily be fatal to her chances because it was possible to negotiate out of them.
"I think she will pick up some vetoes, but the question is how many and how serious they are. I certainly wouldn't be too alarmed if I saw her pick up one or two."
As well as introducing her to the likes of US President Barack Obama, Key also dedicated part of his speech at the UN General Assembly to a pitch for Clark.
Russia has been adamant since day one that it wanted an Eastern European candidate and several candidates are expected to fall to a Russian veto, including possibly Guterres who has been the front runner in every straw poll.
There has been speculation about a late entry from European Commission Vice President Kristalina Georgieva, who is from Bulgaria.
Bulgaria has already nominated Irina Bokova but Bulgaria's Prime Minister has said the government will consider whether to nominate Georgieva after this straw poll.
The current rankings, and the candidates' placings in previous polls are:
1. Antonio Guterres, Portugal (1, 1, 1, 1)
2. Vuk Jeremic, Serbia (4, 2, 3, 3)
3. Miroslav Lajcak, Slovakia (7, 10, 2, 2)
4= Susana Malcorra, Argentina (8, 3, 5, 7)
4= Danilo Turk, Slovenia (2, 4, 8, 6)
6. Irina Bokova, Bulgaria (3, 5, 3, 5)
7 = Srgjan Kerim, Macedonia (5, 6, 6, 4)
7= Helen Clark, New Zealand (6, 7, 7, 8)
9. Natalia Gherman, Moldova (10, 9, 9, 10)
The ratings each candidate got in today's ballot from the choices of "encourage", "discourage", or "no opinion":
• Antonio Guterres, Portugal: 12-2-1
• Vuk Jeremic, Serbia: 8-6-1
• Miroslav Lajcak, Slovakia: 8-7-0
• Susana Malcorra, Argentina: 7 - 7 - 1
• Danilo Turk, Slovenia: 7-7-1
• Irina Bokova, Bulgaria: 6-7-2
• Srgjan Kerim, Macedonia: 6-9-0
• Helen Clark, New Zealand: 6-9-0
• Natalia Gherman, Moldova: 2-11-1
