See full CBS News/New York Times Poll story

Huckabee is now at 21 percent of likely Iowa caucus-goers compared with the leading Republican Mitt Romney, whose support stands at 27 percent. Iowa Republicans cited illegal immigration as the top issue, at 20 percent, with 15 percent saying Iraq was the issue candidates should most discuss. Forty-four percent of GOP caucus-goers believe illegal immigrants should leave the U.S. On the Democratic side, Hillary Clinton, John Edwards and Barack Obama are bunched on top at 25 percent, 23 percent and 22 percent respectively. Clinton and Romney lead comfortably in New Hampshire.