Hayaniya has been the scene of intense fighting since March 25, when Nuri al-Maliki, Iraq's prime minister, ordered a crackdown on militias in the southern port city.
US fighter jets have carried out several air strikes in the district, leaving dozens of people dead.
Sadr City clashes
In Baghdad, police described battles that began during sandstorms on Friday afternoon in Sadr City as among the heaviest in the capital since the crackdown began.
Officials at the Sadr City general hospital said 71 people were admitted for treatment of injuries received in the fighting, as well as 12 bodies.
Lieutenant-Colonel Steven Stover, a US military spokesman, said US troops were involved in sporadic clashes that continued into Saturday.
He said US forces killed two fighters with a helicopter missile strike overnight.
US and Iraqi forces in the area have also come under repeated attack by fighters trying to prevent the construction of a concrete wall through the district.
The Americans are building a wall - up to 3.6m high in some places - along a main street to divide the southern portion of Sadr City from the northern, where Mahdi Army fighters loyal to al-Sadr are concentrated.
Stover said the wall was needed for security reasons - to rebuild a market burned down during fighting - but the project has angered residents, who have been trapped in the crowded battle zone for weeks.
Dr Maha al-Dori, a member of Iraq's parliament loyal to al-Sadr, called the situation in Sadr City a "large humanitarian crisis".
Speaking to Al Jazeera, the MP said: "The hospitals are jammed with dead bodies... The occupation forces completely ban and open fire at any convoy trying to deliver humanitarian aid.
"People here suffer from shortage of food supplies. The occupation forces have burnt the city's markets."
Al-Dori accused Iraqi and US forces of flouting a ceasefire with the Mahdi Army in order to eliminate political opposition, and of targeting women and children in the district.
|A dozen people have been killed in the latest|
fighting in Baghdad's Sadr City [Reuters]
"They have now completely surrounded Sadr City. The media is talking about Gaza, while we now have a second 'Rafah crossing' in Sadr City," she said, referring to Israel's siege of the Palestinian territory.
Clashes between security forces and Shia militias were also reported near Nasiriya, a city about 320km southeast of Baghdad.
Authorities imposed a curfew on the town of Suq al-Shiyoukh after a firefight in which two Mahdi Army fighters were killed and six policemen injured.
In Kirkuk, an Iraqi civilian was killed and three others were wounded in a car-bomb explosion in al-Hajjaj district, police said.
Police also said that a soldier was killed and another injured in a bomb explosion that targeted an Iraqi army patrol in Baiji, west of Kirkuk.
Meanwhile, the US military said an American soldier was killed by a roadside bomb while on patrol in northern Salahuddin province.