The advances were the latest in a fast-paced operation over the past few days that has seen central authorities reclaim the key territory as Kurdish forces retreated
with little resistance.
Thousands of civilians fled Kurdish districts to escape the clashes, heading in buses and cars towards the Kurdistan region.
Around 18,000 have taken shelter in Erbil and Sulaimaniya, the governor of Erbil Nawzad Hadi told reporters.
One of his aides told Reuters the total number of people was about 100,000.
The largely bloodless operation restored to Baghdad's control swathes of territory held by Kurdish forces since 2003, leaving Kurds stunned just weeks after the nationalist fervour
of the independence referendum they held in defiance of the central government.
The Kurds have lost virtually all of the 23,000 square kilometres that they had acquired since 2003, some of it during the chaos of the Islamic State (IS) group's charge across northern Iraq in 2014.
Kirkuk is not one of the three provinces that have been part of the autonomous Kurdish region in Iraq since the US-led invasion of 2003 toppled dictator Saddam Hussein
It is in an area disputed between Baghdad and the Kurds who claim it is theirs historically, since Saddam's forces chased them out and replaced them with Arabs.
Kurds make up two-thirds of Kirkuk's population of 800,000
, with 25 percent Turkmen and the rest Arab Muslims and Christians.