A student said demonstrators were wounded when security forces fired to disperse them in the city south of Baghdad. An officer and a doctor confirmed that people were hit by gunfire but did not say who was responsible.
The demonstration began as Prime Minister Haider al-Abadi visited Wasit University and later moved to the provincial council headquarters when he changed venues, witnesses and an AFP journalist said.
"We received around 70 (injured). Most of them left and three wounded by gunfire and 19 suffering from poisoning as a result of tear gas remain," said Ahmed al-Quraishi, a doctor at a local hospital.
Second Lieutenant Ali al-Sarrai, a member of the security forces tasked with protecting the university, said protesters threw "stones and water bottles and shoes" at Abadi.
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His guards then fired in the air and targeted demonstrators with tear gas, said Sarrai who also confirmed that three people were shot.
The students were protesting against "the lack of services and the spread of corruption in the government", said Ali al-Aboudi, who took part in the demonstration.
Some protesters threw stones at Abadi's convoy, Aboudi said.
Abadi called for "universities to keep away from political conflicts", a statement from his office said.
Video posted on social media showed demonstrators chanting a slogan often used by supporters of populist Shiite cleric Moqtada al-Sadr, who has repeatedly called for protests against corruption in the Iraqi government.
Sadr issued a statement apologising to Abadi and stating that the premier was not personally involved in corruption.
The injuries at the protest in Kut came after seven people -- five demonstrators and two security personnel -- were killed in clashes between Iraqi forces and protesters in central Baghdad on February 11.