Over the course of their war in Iraq, ISIS never actually gained control of the city of Kirkuk, which was seized at the start of the conflict by Kurdish Peshmerga forces. Still, they did seize considerable territory in the southern part of the oil-rich province, and still hold much of it.
Thus, with Iraqi forces entering their seventh month of war in Mosul, and presenting that battle as practically won, despite not gaining meaningful territory in over a month, they are looking past Mosul and toward a big fight against a “regrouping” ISIS in Kirkuk.
Exactly what form this offensive would take will likely depend heavily on how the Mosul battle turns out, with a protracted victory and lingering unrest likely to mean that Kurdish forces would be expected to do more in Kirkuk Province.
There is potential danger to this for the Iraqi government, as any territory they’ve taken from ISIS the Kurds have generally kept, and with officials pledging a secession vote for Kurdish territory pretty much as soon as the ISIS fight is over,any territory lost to the Kurds might be lost permanently.