“We lost everything that day,” Mr. Rodriguez, 48, said. “We had nothing left. And what was left, forget it. It was covered in mold and sewer water.”  Their clothes had absorbed the surge of dirty water like sponges. Water seeped from the bottom of the Sheetrock, past the highest levels of the surge, to the ceiling. Then mold began to grow in thick patches on the walls and furniture.  “And the smell,” Mr. Rodriguez said, scrunching his nose. “That was the first week of the nightmare.”  Nearly four years later, the couple are among thousands who are still displaced after Hurricane Sandy, living in long-term temporary apartments or leaning on friends and family for assistance, exhausted by unending bureaucratic and financial quagmires. Some, like the Rodriguez family, have drained their life savings only to come up short of what they need to restore their flooded homes.  Ms. DeJesus and Mr. Rodriguez are waiting to return home from a basement apartment in Dyker Heights, Brooklyn, which has its own problems with flooding and electrical failures.  “This isn’t our home,” Ms. DeJesus said, shaking her head, in the dining room of that apartment. “We live here day by day, waiting and wondering what’s going to happen, when we can return to our normal lives.”  Eager to move back in the months after the storm, the family tried to fix their home on their own, withdrawing all of their savings, around $25,000, for reconstruction as they waited for a stipend from the Federal Emergency Management Agency.  They started with the living room, where they laid out cots and cooked with a crockpot. Over the next few months, they fixed four of the rooms, including the bedrooms and bathroom. But several months into the project, the couple ran out of money.  Eventually, the couple received $23,000 from FEMA for repairs and replacement items. They received an additional $10,000 from the agency for living expenses. It was not enough. Their renovation expenses totaled more than $55,000. They halted all work on their home; they did not know what to do.  In June 2013, Gov. Andrew M. Cuomo of New York created the Office of Storm Recovery to aid in rebuilding and renovating flood-damaged homes, small businesses and communities across the state. That same month, the city announced the creation of the Build It Back program to help with construction costs. Within five months, some 20,000 families who had lost their homes had applied to the city’s program. Ms. DeJesus and Mr. Rodriguez enrolled in the first month. At last, they thought, their misery would be over.  Two years passed. No help came. Then, in October 2015, Mayor Bill de Blasio promised to overhaul the Build It Back program and return families who had applied to their houses by the end of 2016. “We won’t stop pushing forward until every applicant sees relief,” he said.  The Rodriguez family believes they will still be waiting to go home in the new year.  Elena Trama believes she will be waiting too, mired in a similar agony. She lived in a bungalow community in Sheepshead Bay, Brooklyn, that was under water after the hurricane and without electricity for several weeks. Ms. Trama, 61, who immigrated from Ukraine in 1991, had managed to rebuild and renovate the modest home she had planned to live in for the rest of her life.  “My house is new,” Ms. Trama said. “I built it. I finished it. I was happy.”  But in September 2014, she learned that the Built It Back program could elevate her home above the area’s predicted flood level, potentially protecting it in future storms, something she had not done in her rebuilding efforts. She signed the grant agreements in July 2015 and vacated her home that September so that the work could start. She has waited more than a year without any results.  Phone calls placed by Ms. Trama to Build It Back have yielded only conflicting answers on whether she can move back into her home and live there until the elevation work begins.  The Rodriguez home, which is around 130 years old and includes sections that the family added in the years before the storm, was not structurally sound enough to be elevated. Despite the money and time the family had poured into repairing their home, in 2014 city officials told them that the house needed to be demolished and rebuilt.  Ms. DeJesus and Mr. Rodriguez signed paperwork in January for the work to begin. They moved out of their home and into the Dyker Heights apartment in March.  Now, despite Mr. Rodriguez’s full-time job as a foreman at a burlap supply company, a job he has held for 30 years, the family has fallen behind on their bills. For the first time, they said, they have had to choose which bills to pay, deciding between their monthly mortgage payment of $1,769 and their rent of $2,800 a month. Mr. Rodriguez’s employer has helped by offering loans, he said.  With dwindling resources, the couple turned to Shorefront YM-YWHA of Brighton-Manhattan Beach, a beneficiary of UJA-Federation of New York, one of eight organizations supported by The New York Times Neediest Cases Fund. The Shorefront Y provided the family with $1,631 to help pay their July mortgage payment.  For help with her mortgage payments, Ms. Trama also went to the Shorefront Y, as well as other groups, including New York Disaster Interfaith Services. The Shorefront Y used $1,388 in Neediest funds to cover the entirety of her July payment. Some months, she puts rent on her credit card.  But the Shorefront Y will not be a source of relief for either family in the future. In September, the center’s Disaster Case Management program closed because of a lack of funding. The center continues to help community members with emergency preparedness.  And the blows keep coming. In an October report, the mayor’s office announced that the Build It Back program would not meet its end-of-year target to rebuild all of the single-family homes that were destroyed. According to the report, more than 2,100 homes, or about 25 percent of those eligible, will not be rebuilt by the deadline. The Rodriguez and Trama homes are among those where the work will not be completed.  “Sites are much more complicated than anyone would think from a layperson’s point of view,” Matt Viggiano, a spokesman for Built It Back, said.  Mr. Viggiano cited many reasons for delays, from contractor disputes to housing code violations that must corrected before work begins. In some cases, formidable construction challenges are at play; in Ms. Trama’s bungalow development, for example, homes are very close to one another, adding to the complexity of the work.  Ms. Trama said she had lost faith in a system she once believed would fix the problem. “I thought, ‘It’s good, the city wants to help people,’” Ms. Trama said. “For 25 years, I lived in this country. I was happy. Now I’m broken.”  In limbo, she said she wondered why she had to move from her renovated home if the city was going to wait a year to elevate it. The Rodriguez family also wonders about the taxing delays.  More than four years after the hurricane, their home stands battered, empty and unchanged as a reminder of Hurricane Sandy’s devastation.  “It hasn’t ended,” Ms. DeJesus said. “It never ended. There’s always something else. I go to sleep, and all I see is the mess that is our home. I see the papers and the dollar signs.”  Then, after a pause, she held her hands in the air. “I want to wake up.”  Donations to the Neediest Cases may also be made with a check or over the phone.  A version of this article appears in print on November 20, 2016, on page A19 of the New York edition with the headline: 4 Years After Hurricane Sandy, 2 Households Still Wait to Return Home. Order Reprints| Today's Paper|Subscribe
Despite federal, state and city programs created to help rebuild damaged homes, the Rodriguez family and Elena Trama of Brooklyn