Four Palestinians were killed by Israeli fire on Friday, bringing the total over four weeks of protests to 38.
Israel has pledged to stop damage to the fence, infiltrations and attacks, and alleges there have been attempts at all three.
For protests set for Friday, Gazans were hoping to send dozens of kites over the fence, including some carrying notes telling Israelis "there is no place for you in Palestine".
The individual as been identified as Mohammad Ibrahim Ayoub, he was said to have been killed by live ammunition during the fourth consecutive week of clashes after Palestinian factions announced a six week "Great March of Return".
Health officials said that two Palestinians were killed by army fire near the border Friday. Some of the injured were overcome by tear gas, hit with rubber-coated steel pellets or hit by shrapnel.
Avigdor Lieberman was visiting near the Gaza border Friday when asked if he was anxious.
"I came here without a flak jacket", Lieberman noted during his border tour.
"You are strongly advised not to participate in acts of violence against Israeli army troops or Israeli citizens", the flyers read, according to the newspaper.
The clip showed the group's operatives looking through the scope of a rifle at Israeli military officers.
Israel is closely guarding the fence and warns it will not tolerate attempts to breach it.
Palestinian protesters fly a kite with a burning rag dangling from its tail to during a protest at the Gaza Strip's border with Israel on Friday.
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The ministry says 33 of those injured Friday were taken to hospitals and the rest were treated in field clinics.
Israeli forces were using tear gas in addition to live fire. Troops responded "with riot dispersal means and are firing in accordance with the rules of engagement", it said.
The participation of large numbers of young Palestinians in protests that involve the strong likelihood of being shot dead or grievously wounded is more broadly a testament to the desperate conditions they confront within the blockaded and besieged enclave of Gaza, which has been justifiably compared to an open-air prison.
The atmosphere near Gaza City was festive Friday, even as protesters clashed with Israeli troops at the fence. He said one of the latest victims was a 25-year-old disabled man.
The Israeli military had no immediate comment. Peace talks between Israel and the Palestinians have been stalled for several years and Israeli settlements in the occupied territories have expanded.
The military told Channel 10 news it was investigating the video and the incident as a whole.
The protest began on March 30, and has seen tent encampments spring up near the Israeli-imposed restricted zone along the 40km border fence. Several hundred people have also been wounded by Israeli sharpshooters.
By late afternoon, crowds had swelled and smoke from burning tires billowed over the protests.
The leaflets were dropped on Friday, ahead of what's to be the fourth large-scale protest, largely organized by Gaza's rulers from the Islamic militant Hamas, in part to challenge a decade-old border blockade of the territory.
White House envoy Jason Greenblatt, a member of President Donald Trump's Mideast team, said on social media that Palestinians in Gaza have a "right to protest their dire humanitarian circumstances."Organizers "should focus on that message, not stoke the potential for more violence with firebombs and flaming kites, and must keep a safe distance from the border", said Greenblatt, adding, "The cost of these demonstrations is too high in loss of life and injuries", as reported by the Associated Press (AP)".
"If Hamas decided not to participate, the number of protesters would be really low, " he said.