She said: "I think its really important that we are clear that Abdulla's father was the deputy agriculture minister and therefore was a member of the Hamas-run government which is different to being part of the military wing of Hamas.
"I think externally it is often simplified that he was in Hamas and I think it is an important point of detail that we need to continually remind people of the difference."
Hamas is proscribed in its entirety as a terrorist organisation in the UK and being a member of either the military or political wing of the group is a criminal offence.
The Board of Deputies has issued a strong condemnation of Turness' remarks, which it claimed contradicted the legal definition of Hamas as a proscribed group.
Vice President for Security, Resilience and Cohesion Andrew Gilbert said: “We are extremely concerned that within days of the BBC’s damning report into their flawed documentary Gaza: How to Survive a Warzone, the CEO of news, Deborah Turness, appears to be obfuscating and minimising the BBC’s failings.
“Hamas is a proscribed terrorist organisation, and this so-called distinction between political and military wings has been categorically dismissed as artificial by the British government. Her remarks suggest that the BBC has not learned even basic lessons from their recent mistakes. Errors of this kind must come with accountability.
“The Board of Deputies will continue our engagement with regulators, government and BBC executives to ensure accountability, and the urgently needed changes within the corporation.”