Medics at al-Awda Hospital in the central Gaza Strip said at least three people were killed and dozens wounded by Israeli fire as they tried to approach a GHF site near the Netzarim corridor.
Two others were killed en route to another aid site in Rafah in the south.
An air strike killed seven other people in Beit Lahiya town north of the enclave, medics said.
In Nuseirat camp in central Gaza Strip, medics said an Israeli air strike killed at least 11 people in a house.
The rest were killed in separate air strikes in the southern Gaza Strip, they added.
There was no immediate comment from the Israeli military.
The GHF began distributing food packages in the enclave at the end of May after Israel partially lifted a near three-month total blockade.
Scores of Palestinians have been killed in near-daily mass shootings trying to reach the food.
The United Nations rejects the Israeli-backed new distribution system as inadequate, dangerous and a violation of humanitarian impartiality principles.
Later on Sunday, COGAT, the Israeli military aid co-ordination agency, said that this week it had facilitated the entry of 292 trucks with humanitarian aid from the UN and the international community, including food and flour, into the Gaza Strip.
It said the Israeli military would continue to permit the entry of humanitarian aid into the enclave while ensuring it did not reach Hamas.
Hamas denies Israeli accusations that it steals aid and says Israel is using hunger as a weapon against the residents of the Gaza Strip.
The Gaza health ministry said in a statement on Saturday that at least 300 people have so far been killed, and more than 2600 wounded, near aid distribution sites since the GHF began operations in the strip.
"These are not humanitarian aid, these are traps for the poor and the hungry under the watch of occupation planes," said Munir Al-Bursh, director-general of the health ministry.
"Aid distributed under fire isn't aid, it is humiliation," Bursh posted on X on Sunday.
The war in the Gaza Strip erupted 20 months ago after Hamas-led militants raided Israel and took 251 hostages and killed 1200 people, most of them civilians, on October 7, 2023, Israel's single deadliest day.
Israel's military campaign since has killed nearly 55,000 Palestinians, most of them civilians, according to health authorities in Gaza, and flattened much of the densely populated strip, which is home to more than two million people.
Most of the population is displaced, and malnutrition is widespread.
