Derry City and Strabane District Council received more than of 5,500 applications for its emergency heating hardship fund.
The £258,000 support scheme was only designed to help 2,405 households “in emergency situations” in the district.
The council said 1,600 eligible referrals had been confirmed as of Wednesday, December 22, and payments had started.
“It shows the depth of need out there,” said Sinead Quinn, from Derry Against Fuel Poverty.
People eligible for the scheme will receive a one-off payment of up to £100, paid directly to their energy providers.
The funding is only available to oil and electricity customers in the council area and not to gas customers.
Applications for the scheme went live earlier in December after a series of delays.
The council website began accepting referrals at 09:30 GMT on 6 December, but closed at 16:00 the same day after receiving thousands of applications.
All applicants to the programme will be advised this week of their eligibility status, the council have confirmed.
“People out there are clearly scared about how they are going to afford to stay warm in their homes this winter,” Ms Quinn said.
Ms Quinn said her organisation has received calls from people desperate for support and said she was worried the roll out of the scheme is too slow.
“I know council are doing everything they can, but we have people who are desperate right now,” she said.
A council spokesperson said it was working with its partners to progress with the referrals and payment process as soon as possible.
The council said the first tranche of 780 referrals were submitted to its fuel payment partner last week and the second tranche of 790 referrals were submitted on Tuesday.
It said it was preparing the third tranche of eligible referrals, which it said was expected to be about 500 referrals.
Earlier in the week, the UK government confirmed households across the North of Ireland would receive a single payment of £600 in January to help with their energy bills.
The scheme will combine two different payments.
Firstly, a £400 payment that was promised by the government to all households in the UK to help with rising energy bills.
A second £200 payment, which was announced in November to address the high number of home heating oil users in the North of Ireland.
This payment is being given to all households regardless of whether they use oil.
The council said the introduction and roll out of the scheme had been a challenging process.
It said referrals were submitted electronically, scanned and also delivered by hand and said that “there were many duplicate submissions received” across all eight district electoral areas.
It said that due to “the significant demand for this programme” and “limited amount of funding available”, the scheme was “not able to support everyone who is in need given the scale of the current crisis”.
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