In September, Mr McCully warned the UN in New York that it was losing credibility over its inability to act in Syria, where 25,000 people had been killed in a civil war.
New Zealand is seeking a non-permanent seat on the Security Council for the 2015-16 term.
Miss Clark also spoke about the risks in failing to co-ordinate a global response to climate change.
"It would be a tragedy for future generations if today's leaders and decision-makers prove incapable of taking the bold decisions which are necessary to stop catastrophic and irreversible change to the world's climate."
She said there was limited accountability for the agreements that had been reached on carbon emissions, and no meaningful consequences for failing to reach reduction targets.
There was also concern about the fragmentation of global climate change agreements and negotiations.
"Critics of this fragmentation have argued that agreements reached by only some countries are inherently flawed."
Last week, New Zealand opted out of committing to the second phase of the Kyoto Protocol, which sets legally binding targets for reducing greenhouse gases.
New Zealand would instead join the United States, Japan, China and other large economies in committing to the UN Framework, which was not legally binding.