Kurdish-led forces attacked the provincial capital, also called Raqqa, a week ago, hoping to drive the militants out with the aid of US-led coalition airstrikes.
The Britain-based Syrian Observatory for Human Rights said IS launched counterattacks in one of the neighborhoods where the Syrian Democratic Forces are advancing. The SDF said it is helping to get people to safety in at least four neighborhoods seized from IS militants.
UNHCR spokesman Andrej Mahecic said the barriers to movement have made aid operations "costly and complex."
He said all land routes to the region have been blocked by other parties to Syria's civil war that are hostile to the US-backed force, forcing the aid agency to rely on airlifts.
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"Resources are also badly needed," said Mahecic. "Funding is not keeping up with needs on the ground."
The UN has managed to raise only USD 29 million of the USD 153 million it budgeted to meet humanitarian needs in Raqqa province.
Raqqa Is Being Slaughtered Silently, an activist group reporting from the city, says food and medicine are running low and shops are shut.
Human Rights Watch has meanwhile called on the US-led coalition to make protection of civilians a priority in the campaign to recapture Raqqa.
The New York-based group said in a statement that the United States and allied ground forces must respect the human rights of everyone caught up in the battle.
It also urged the US to investigate airstrikes that have allegedly targeted civilians, respect detainee rights, provide safe passage for the displaced and intensify efforts to clear land mines. And it sought guarantees against enlisting child soldiers into the ranks of US partner forces.
HRW reported in 2014 the SDF's leading faction, the People's Protection Units (YPG), had enlisted soldiers under the age of 18.