Under the new tax law that was passed in 2017, there is no deduction on the federal tax return for moving expenses for years 2018 through 2025. (There is an exception for active-duty military members who are moving because of a permanent change of station, but that does not appear to apply to you.)
Your employer was correct to include the partial reimbursement in your W-2. It is just additional wages.
On your New York tax return you can exclude the employer's reimbursement from your New York income, and deduct unreimbursed moving expenses.
