“The government’s responsibility is to watch out, to be custodians of the proper and effective use of those funds,” he said. “I don’t understand why they need to pay hundreds of millions of those dollars to middlemen who turn around and permit overpricing.”

The pricing issues and widespread complaints of long waits and shoddy work highlight the challenges of managing a billion-dollar disaster aid program in a region that is far from the mainland, with institutions that historically have had limited outside oversight or accountability.

Puerto Rico housing officials said they were proud of the repair program, and that prices were in many cases less than those paid in other disasters, including repairs after Hurricanes Irma and Maria hit the Virgin Islands, which have similar transport challenges.

Michael Byrne, FEMA’s federal coordinating officer for Puerto Rico, said the housing department had done an impressive job of getting homes repaired quickly for people who had nowhere else to turn.

“By the end of November, I fully expect them to have repaired about 120,000 homes,” Mr. Byrne said. “That’s pretty impressive.”

Records show a large gap between the amounts FEMA contractors hired by the Department of Housing were paid and the actual cost of the work that was ultimately performed. Across the board, from removing debris and cleaning mold to repairing roofs and installing appliances, the amounts for labor and materials that were paid to the people who actually performed the work were only about 40 percent of what FEMA was assessed, meaning homeowners got less help than many of them expected.

In case after case, a door worth about $50 would be billed to FEMA at perhaps $700, with a succession of intermediary contractors passing along costs and profits along the way, according to María Elena Villalobos, who worked as both an inspector and an administrator for several companies in the housing repair program. “A lot of the money went down the drain,” Ms. Villalobos said.