The United States opposed the independence initiative, which the Kurdish authorities say passed overwhelmingly. Washington said the vote would destabilize Iraq and undermine the American-led coalition battling Islamic State militants.

The fighting on Tuesday seemed to slow — at least for a day — Baghdad’s effort to impose authority over remaining contested areas and border crossings with Turkey, Iran and Syria.

Kurdish commanders, whose forces are outnumbered and outgunned, said late Tuesday that Iraqi reinforcements were arriving at the scene of one clash, in the town of Makhmour, about 70 miles southeast of Mosul, and that they anticipated a government assault at any time.

Jabar Yawar, secretary general of the pesh merga, said in a telephone interview that Iraqi troops moved without warning toward Kurdish positions in Makhmour on Tuesday morning. The move “led to confusion and then to a clash,” Mr. Yawar said.

Mr. Yawar said there were government casualties, but he had no details.

Hemin Hawrami, a senior aide to Mr. Barzani, wrote in a Twitter post that pesh merga fighters destroyed two government Humvees. Mr. Hawrami wrote in a separate post that the Kurds had destroyed two armored personnel carriers and a Humvee while repulsing a government attack outside Rabia, near a border crossing with Syria.

Those accounts could not be confirmed. The Iraqi military command, which provided regular updates of the operation in and around Kirkuk last week, did not respond to several requests for comment Tuesday.

In a statement Tuesday, the pesh merga command accused Baghdad of “bad intentions” and promised to resist any further attacks. “The pesh merga will defend the land of Kurdistan and the life of the people with all their strength,” the statement said.