Combinatorial relations among relations of $C_2\text{sp}(1)$-standard modules for higher levels

For an affine Lie algebra $\widehat{\mathfrak{g}}$ the coefficients of certain vertex operators which annihilate level $k$ standard $\widehat{\mathfrak{g}}$-modules are the defining relations for level $k$ standard modules. In the paper \cite{PS3} combinatorial structure of the leading terms of mentioned relations for level $k=2$ standard $\widehat{\mathfrak{g}}$-modules and affine Lie algebras of type $C_n\text{sp}(1)$ is given. The main result of this paper is a construction of combinatorially parameterized relations among relations of $C_2\text{sp}(1)$-standard modules for all levels.