Cairo (CNN) -- Protesters marching on Egypt's presidential palace Tuesday night broke through barbed wire around the building and hurled chairs and rocks at retreating police.
Officers lobbed tear gas back at them.
After initial clashes, police drew behind fences and protests were peaceful for several hours.
More violence broke out at the headquarters of the Freedom and Justice Party in Menia, south of Cairo. At least 19 protesters were injured, according to Mahmoud Amin, who is in charge of ambulance operations outside the palace.
The Freedom and Justice Party -- an Islamist political group that has links to the Muslim Brotherhood -- is the party of President Mohamed Morsy.
Party head Dr. Hussein Sultan said the front of the headquarters is damaged, and at least one protester fired a shot in the air.
Egypt's Health Ministry said at least 50 ambulances had been dispatched to locations around Cairo, including 20 to the presidential palace. Hospitals were placed on high alert in anticipation of injured protesters, the ministry said.
Many in Egypt believe a new draft constitution in the country, which will be put to a popular vote on December 15, is unfair in its wording, and is an attempt by Morsy to grab more power.
Morsy was not at the palace when the protesters descended, his office said. He was out meeting with government officials, they said.
The protests cap a day of silent protest by media organizations opposing the country's new draft constitution and an edict Morsy issued nearly two weeks ago to expand his powers.
They feel the constitution does not sufficiently protect freedom of the press and, on Tuesday, a dozen partisan and privately-owned papers were not on the newsstands in protest.
Four Egyptian satellite channels are expected to go off the air on Wednesday in solidarity.
Freedom and Justice Party media adviser Murad Ali warned that organizers of the protest outside the palace could face consequences.
They "must bear the responsibility of the demonstrations they called, and bear full responsibility for any violence that may result from poor organization," he said.
Journalist Sarah Sirgany in Cairo and CNN's Ben Brumfield in Atlanta contributed to this report.