Combinatorial relations among relations for level 2 standard $C_n\text{sp}(1)$-modules

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 this paper we study a combinatorial structure of the leading terms of these relations for level $k=2$ standard $\widehat{\mathfrak{g}}$-modules for affine Lie algebras of type $C_n\text{sp}(1)$ and the main result is a construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will play a key role in a construction of Groebner-like basis of the maximal ideal of the universal vertex operator algebra $V_{\mathfrak{g}}^k$ for $k=2$.