Birth registration ensures children born to Syrian parents in Lebanon will eventually be able to claim a Syrian identity card and benefits.
The camp, which lies near the country's southern border with Jordan, houses thousands of Syrians in dire living conditions.
The aid had been intended for delivery to families in the city of Taez in November 2018, but was detained at a checkpoint for many months.
Recent events have made it clear that Iran and the Houthis seek to prolong the conflict in Yemen to further their own goals, experts say.
Caution prevails in the southern port city in the aftermath of clashes between Yemeni forces and the Southern Transitional Council.
UNICEF has been working to improve conditions and provide services for thousands of children living at the al-Hasakeh camp.
According to UNICEF, the salaries of more than 1.25 million public sector employees have been suspended for more than two years.
Yemeni and international activists are calling on the Iran-backed militia to overturn the death sentences it handed to 30 alleged spies.
Some Syrian refugees who opted to return to their homeland have been arrested and imprisoned, in breach of reconciliation agreements.
The Houthis are diverting aid to NGOs set up by the militia and are selling food aid in the black market, according to Yemeni and UN officials.