A man believed to belong to Islamic State blew himself up inside the Sayidati al-Jamila ("My Fair Lady") restaurant at lunchtime, killing at least four people and wounding 15.
The second attack was a suicide car-bomb that killed a soldier and wounded four others in the al-Nour district.
Both districts were recently retaken from the militant group by U.S.-backed Iraqi forces. An explosive belt of the style usually worn by suicide bombers also blew up in a street of al-Zuhour without causing casualties.
The offensive that started in October to capture Mosul from Islamic State paused last month after Iraqi forces took all the districts lying east of the Tigris river that bisects the city.
Friday's attacks are the second major incident in districts retaken by Iraqi forces. A triple car bombing killed at least 23 people in the eastern suburb of Kokjali on Dec. 22.
Reporting by Maher Chmaytelli; Editing by Janet Lawrence
