This payment was a "signing bonus", an incentive for you to accept the position - it was properly reported on Form 1099-NEC and is subject to self-employment tax and income tax. The full amount is "NY source income", since the recruiting agency and the employer are both located in NY.
The bonus was not from sporadic activity - it is directly related to your main job (you wouldn't have gotten it otherwise).
From The Tax Effects of Paying Bonuses to Employees: Employee bonuses are always taxable to employees as an employee benefit, no matter how or when they are paid. For example, a bonus paid to an employee at the time of hire (sometimes called a "signing bonus") is subject to all employment taxes. The employees must pay federal and state income taxes and FICA taxes (Social Security and Medicare) on bonus pay.
Even though the payment did not come from your new employer, it is reasonable to conclude the recruiter was acting on behalf of the employer, ie. as as an agent of the employer.
Don't worry about how it transferred to your NJ return - it will be treated the same way for NJ state tax purposes as if it had transferred as a 1099-NEC.
