  We study the quantum vertex algebraic framework for the Yangians of RTT-type
and the braided Yangians associated with Hecke symmetries, introduced by
Gurevich and Saponov. First, we construct several families of modules for the
aforementioned Yangian-like algebras which, in the RTT-type case, lead to a
certain $h$-adic quantum vertex algebra $\mathcal{V}_c (R)$ via the
Etingof-Kazhdan construction, while, in the braided case, they produce
($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Next, we show that the
coefficients of suitably defined quantum determinant can be used to obtain
central elements of $\mathcal{V}_c (R)$, as well as the invariants of such
($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Finally, we investigate a
certain algebra which is closely connected with the representation theory of
$\mathcal{V}_c (R)$.
