New York City spent more than $1 billion last year on private-school tuition reimbursements and related expenses for students with disabilities, a growing cost that Mayor Zohran Mamdani’s administration says it wants to reduce by expanding services in public schools. The administration is adding specialized programs and staff to address delays in student evaluations, while investing in preschool special education. Mamdani has projected that reducing spending on these cases and changing how they are managed could save the city $149 million a year. The cases, often called Carter cases, arise when families seek funding for services outside the public school system. Awards may cover private-school tuition, educational services, transportation or legal costs. Families can pursue the claims through a due process system. City spending on the claims rose from $47 million in 2005 to $312 million in fiscal year 2016, before exceeding $1 billion last year. The rising total puts pressure on a city budget already facing chronic deficits, while underscoring the stakes of ensuring students can get needed support in public schools. The administration’s approach rests on the idea that stronger public-school programs and quicker evaluations can reduce the need for families to seek private placements or pursue claims. Whether expanded capacity will deliver the projected savings remains to be seen; the city has not yet demonstrated that the changes will reverse the spending trend.