  We apply the theory of $\phi$-coordinated modules, developed by H.-S. Li, to
the Etingof--Kazhdan quantum affine vertex algebra associated with the
trigonometric $R$-matrix of type $A$. We prove, for a certain associate $\phi$
of the one-dimensional additive formal group, that any $\phi$-coordinated
module for the level $c\in\mathbb{C}$ quantum affine vertex algebra is
naturally equipped with a structure of restricted level $c$ module for the
quantum affine algebra in type $A$ and vice versa. Moreover, we show that any
$\phi$-coordinated module is irreducible with respect to the action of the
quantum affine vertex algebra if and only if it is irreducible with respect to
the corresponding action of the quantum affine algebra. In the end, we discuss
relation between the centers of the quantum affine algebra and the quantum
affine vertex algebra.
