Some analysts in Israel have pointed to the hunger strike as a bid by Barghouti to show Abbas he still wields political power.
Around 1,500 Palestinians in Israeli jails have been refusing food since April 17.
The Palestinian "martyrs' fund" pays about 35,000 families of Palestinians killed and wounded in the long-running conflict with Israel, and says the money amounts to welfare payments to victims.
The head of the Palestinian Prisoners' Club, Qadoura Fares, called the video a fabrication, stressing Barghouti was being held in solitary confinement at Kishon prison and had no access to food. His wife and other Palestinian officials have dismissed it as fake and as an attempt by Israeli authorities to break the morale of other hunger-striking prisoners. Those still shots were taken with a hidden camera through a hole in the wall of his cell. Hamas immediately became a major rival of Fatah, the largest Palestinian faction, building grassroots support among conservatives mainly in Gaza but also in the West Bank.More news: Wife of American detained in North Korea pleads for his release
Haniya, seen as a pragmatist within Hamas, was elected by its Shura Council on Saturday.
Haniya, seen as a pragmatist within the movement, is expected to remain in the Gaza Strip, the Palestinian enclave run by Hamas since 2007.
Hamas' shift to Gaza comes at a time of growing financial pressure on the territory, ruled by the Islamic militant group since it drove out forces loyal to Western-backed Palestinian President Mahmoud Abbas in 2007.
Since quitting his longtime base in Damascus in 2012, Mashaal has mostly lived in lavish suites in the capital of the oil rich gulf state of Qatar.More news: Fox harassment claims create clouds around Sky takeover bid
Haniya takes charge of Hamas as it seeks to ease its global isolation while simultaneously marginalising hardliners within the movement.
The Israeli prime minister threw Palestinian group Hamas' new political platform in a rubbish bin in a video uploaded to his social media account.
Islamic Jihad's deputy leader, Ziad al-Nakhala, said he considered the new Hamas platform as a progress towards a "deadlock", adding that his movement does not accept a "Palestinian State within the 1967 borders because such a concession damages our aims". The new charter also omits an odious claim in the 1988 document that Muslims are called upon by God to kill Jews.
Abbas made the announcement on taxes the same day that Ismail Haniyeh, who was born and grew up in the Shati refugee camp, was elected leader of the Hamas politburo, following Khaled Meshal who served two terms. In a prosaic repetition of history, Hamas - the 30-year-old, Gaza-based Islamic resistance movement - chose to amend its charter, releasing a political document of principles that basically severs ties with the Muslim Brotherhood and accepts a Palestinian state on the 1967 borders, with Jerusalem as its capital, without recognizing Israel.More news: Portion of tunnel containing nuke waste collapses at Hanford
Haniya was one of Yassin's closest Hamas leaders, and lived with the quadriplegic leader, guiding him in his wheel chair and spending most of his time with him whether feeding and bathing him or reading for him.