Dozens of youths threw rocks at the Ettihadiya palace after a peaceful march by thousands of demonstrators who accused Mursi's conservative Muslim Brotherhood of hijacking Egypt's democratic revolution and seeking to monopolise power.
Police responded by firing water cannon and teargas from the walls of the presidential compound, which have been raised in some places and shielded by barbed wire after petrol bombs set fire to a building in the grounds last week.
Riot police later emerged to chase the protesters away from the palace and into side-streets.
The clashes, which appeared smaller and less violent than previous bouts of anti-Mursi unrest, were broadcast live on some television channels.
"We may be few in numbers but we will not back down from fighting criminals dressed up in police uniform," said Ahmed Farghaly, a protester outside the presidential palace.
Earlier, anti-Mursi protesters briefly occupied one of the main bridges across the Nile, disrupting rush-hour traffic in the heart of the capital.
The main Cairo rally was in the central Tahrir Square, focal point of three weeks of demonstrations in 2011 that led to Mubarak's resignation.
Amid opposition calls for Mursi to sack his cabinet and form a national unity government, Prime Minister Hisham Kandil said in a government statement: "The revolution will bear fruit through serious work and effort, and by avoiding incitement and political brinkmanship."
Mubarak, who ruled Egypt with an iron fist for 30 years, was sentenced to life imprisonment last June for his role in killing protesters after a trial seen as setting a precedent for holding Middle East autocrats to account.
Now opposition parties are demanding that Mursi be put on trial over the deaths of nearly 60 demonstrators in anti-government protests that erupted on January 25, but the public prosecutor says there is no evidence to link the democratically elected president with the deaths.
