How to attribute moving expense reimbursement on state returns?

I moved from Canada to New Jersey for a job in New York. My employer reimbursed $4000 and withheld taxes from the repayment to me.


My employer is located in NY and reported the $4000 as NY state income on my W-2. My employer told me that tax law dictates that box 1 and box 16 on my W-2 need to match, even though New York does not tax moving expense reimbursement. They said that my “true” NYS income when filing is $4000 less than what is reported on my W-2. When I use turbotax to fill my return it automatically pulls box 16 from my W-2. Do I need to reduce this amount by $4000 when filing?

 

Furthermore, I moved to NJ not NY. So I am under the impression that the $4000 would not be allocated to NYS anyways. 

Do I reduce my NYS income by $4000 when filing?