The fighting between police and armed men in west Mosul, including mortar rounds fired at checkpoints, killed nine police and wounded seven, police and a doctor said.
Four gunmen were also killed in the clashes.
In Rashidiyah, north of Baghdad, the car bomb exploded as worshippers left Friday prayers at Al-Ghufran mosque, killing at least four people and wounding 22, an interior ministry official and a medical source said.
And in Al-Amil in south Baghdad, a magnetic "sticky bomb" wounded a police captain, while a roadside bomb wounded three more police in Taji, north of the capital, the ministry official and medical sources said.
According to a UN report, April was the deadliest month for Iraq in almost five years.
"The month of April was the deadliest since June 2008. A total of 712 people were killed and another 1,633 were wounded in acts of terrorism and acts of violence," a statement from the UN mission in Iraq said on Thursday.
A wave of violence began on April 23 when militants invaded security checkpoints near the town of Hawijah in north Iraq, sparking clashes in which 53 people were killed.