Hadi appealed to the UN Security Council on Tuesday to "shoulder its responsibilities.
The Huthis took control of Al-Anad airbase following "limited clashes" with forces loyal to Hadi, an official said.
Yemen is increasingly divided between a north controlled by the Huthis, allegedly backed by Iran, and a south dominated by Hadi supporters.
The UN Security Council, Western countries and Gulf Arab monarchies have backed Hadi as the country's legitimate ruler.
In sentencing Mr Justice Sweeney, sitting at the Old Bailey, told Hadi that "justice could be tempered by mercy" particularly in cases involving someone who is vulnerable or in a relationship with a defendant.
The judge told Hadi that there had been "an element of choice'' in her decision not to alert the authorities after Dart revealed his intentions to her in November 2011.
The judge noted that Hadi was of good character and that neither she nor any member of her family had shown any Jihadist sympathies.