The images of the higher generators via the evaluation map for the affine Yangian of type $A$

The affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$ has several presentations: the current presentation, the minimalistic presentation and so on. The evaluation map for the affine Yangian was given by using the minimalistic presentation. One of the issues about the evaluation map is that the images of the evaluation maps are unkown except on finitely many generators. In this article, we write down the images of the higher generators of the current presentation via the evaluation map for the affine Yangian of type $A$ explicitly.