Two police officers, one member of the Brigades and a bystander were wounded in one clash, the Palestinian Centre for Human Rights said.
In the second clash, which took place at the Rafah terminal, two bystanders were wounded.
Elsewhere, armed men, mainly belonging to al-Aqsa Martyrs Brigades, briefly stormed a military hospital in southern Gaza and a power plant overnight, demanding unpaid wages and jobs, the centre said.
Earlier on Monday, Brigades fighters blocked the main road leading to the Erez terminal, the crossing point into Israel used mainly by Palestinian officials.
The motorcade of the Palestinian president, Mahmoud Abbas, was on it way to Tel Aviv at the time for a follow-up meeting on the humanitarian situation in Gaza.
The fighters later exchanged fire with the Palestinian Authority's security forces at the Ministry of Foreign Affairs in Gaza City and shot at a police station.
On Sunday, municipal workers went on strike because they have not been paid, and uncollected trash piled up throughout the city.
Last week, taxi drivers blocked roads and went on strike over rising fuel costs.
Palestinian police trade fire with
armed men in Gaza
The cash-strapped PA is increasingly finding it difficult to pay the salaries of the 140,000 people it employs, among them 70,000 security forces, in the face of rising international pressure and economic isolation of the planned Hamas government.
Israel has vowed to withhold tax revenues amounting to $52 million a month owed to the Palestinian Authority, and the US has demanded the PA return an equal amount of money in foreign aid.
Israel briefly reopened the main commercial crossing into Gaza on Monday to allow trucks to deliver food following warnings of an impending humanitarian crisis in the Palestinian area, Palestinian police said.
The Karni (al-Mintar) crossing was partly reopened for goods leaving Israel destined for the Gaza Strip shortly after 1pm (1100 GMT), journalists said on Monday.
The checkpoint in central Gaza was shut less than an hour later. A Palestinian security official told AFP that Israeli officials had ordered the closure because of an attack alert.
A Palestinian walks past trucks
after the Karni crossing opened
Salim Abu Safieh, director of the Palestinian-staffed side of Karni, told AFP the Israeli authorities had committed themselves to opening the Karni crossing eight hours a day.
Earlier on Monday, a spokesman for the Defence Ministry in Tel Aviv said the crossing, which has been closed for most of the year, was being reopened on a temporary basis to allow aid to be brought into the territory.
"We have decided to partially reopen Karni for a limited time for humanitarian reasons," the spokesman told AFP.
The Karni crossing is Gaza's commercial lifeline, the only point through which large-scale import of wheat and other goods can take place.
The step was taken after talks on Sunday hosted by Richard Jones, the US ambassador in Tel Aviv, where the Palestinians pressed for the reopening of Karni.
The UN agency for Palestinian refugees said on Sunday that Gazans were running short of basic foodstuffs, including bread and sugar.
Israel had previously offered to channel aid through another border crossing in the south of the territory, known as Kerem Shalom, but the idea had been resisted by the Palestinians, who insisted that Karni should be reopened.
Bakeries have run out of wheat
flour. Photo: Laila el-Haddad
Kerem Shalom is inside Israel, at the corner of the border with Gaza and Egypt.
Israeli authorities say the Karni crossing has been closed because of security threats, specifically, fears that tunnels had been built under the crossing.
Palestinians have rejected such claims, saying the closure is politically motivated.
Gaza's 1.5 million Palestinians face an unprecedented food shortage because of Israeli closures that have prevented importing wheat, the United Nations Office for the Coordination of Humanitarian Affairs in the Occupied Palestinian Territories (OCHA) said on Sunday.