It is tough to say which should come first, but fighting poverty is a more realistic problem that is in the now and needs to be handled before things get worse and everyone ends up broke and living a hard life. Climate change should be addressed and a goal they work towards, just not as pressing.
Poverty is more apparent then climate change and most people know that poverty is a problem and not a lot accept the theory of climate change. Plus there is a large majority of people that live in poverty and most or all of them would prefer them to be helped over climate control.
Climate change is happening and has been since the start of the industrial revolution, you pump shed loads of carbon into the atmosphere it starts feeding into the natural carbon cycles which are naturally saturated, and at present are less able to compete with co2 increases due to deforestation and soil errosion etc. much of the carbon is absorbed into the oceans which increases acidity and impacts on marine ecosystem greatly ( ecosystems that are relied upon by more than half the worlds population). Some of the carbon goes into the atmosphere, its been known to rtain radient heat since the time of svante arrhenius and john tyndall in the late 19th century. That latent heat warms the atmosphere and the oceans causing changes to oceans currents and jet streams. The temperature of the earth has increase almost a degree on average in 100 years might not seem much, but some places are warming much faster like the arctic etc, in many chemical systems upsetting condition by as little as this can have huge implications. These temperature increases are facilitating massive changes in local climates and has increased extreme whether like major droughts (africa, china, USA brazil) and flash floods (india, bangladesh china southeast asia) and where do the most impoverished people live ? --- china, southeast asia, sub-saharan africa, west africa, south america. Climate change is real and there is more evidence to support it than not. And these countries are the most at risk, droughts and flash floods can completely wipe out crop production in places like this.
I don't think its right to push either issue aside when they are equally as important as each other. Prevention is better than cure in my opinion.