Joshua Boyle and his wife Caitlan Coleman have been held by a Taliban-linked militia since 2012 after they were kidnapped near Kabul, Afghanistan.
In the video, the family describe the hell they have been living in for the past four years and ask for help and to be freed.
Sitting in the video with their two young sons, who were both born in captivity, the mother and father call on Barack Obama and Donald Trump for their help.
The couple were kidnapped in Afghanistan while they were backpacking through central Asia.
Speaking in the clip, Coleman who wears a black robe, says the young children have witnessed her being abused by their captors.
She said: “We understand both sides hate us, and are content to leave us and our two surviving children in these problems.
“But we can only ask and pray that somebody will recognise the atrocities these men carry out against us as so-called retaliation, in their ingratitude and hypocrisy.
“My children have seen their mother defiled.”
In the clip the children both sit on their father's lap as their mother reads from a prepared script asking for help.
Coleman was five months pregnant when she was kidnapped and gave birth while imprisoned, before their second son was born in 2015.
The Haqqani network has a strong following in both Afghanistan and Pakistan, and is closely interlinked with the Taliban.
Coleman continued: “Obama, your legacy on leaving office is probably important to you, and our lives and those of our children are to us.
“Just give the offenders something so they and you can save face and we can leave the region permanently.”
The couple have appeared in other videos, however this is the first time their children have been shown on camera.
Boyle, 33, and Coleman, 31, were in a video released four months ago, which warned that the couple would be killed if the Afghan government did not stop executing Taliban prisoners.
Speaking to new President-elect Donald Trump, Coleman said: “They want money, power, friends. You must give them these things before progress can be made.
“A five-year hostage-taking is too long and indicates failure on every side.”
The video, which is believed to have been filmed on December 3, and is being reviewed by US and Canadian intelligence agencies.
Posted by Women Against Shariah on Tuesday, December 20, 2016