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Let $\mathscr{A}_q$ be the $K$-theoretic Coulomb branch of a $3d$$\mathcal{N}=4$ quiver gauge theory with quiver $\Gamma$, and $\mathscr{A}'_q\subseteq \mathscr{A}_q$ be the subalgebra generated by the equivariant$K$-theory of a point together with the dressed minuscule monopole operators$M_{\varpi_{i,1},f}$ and $M_{\varpi^*_{i,1},f}$. In this paper, we construct anassociated cluster algebra quiver $\mathcal{Q}_\Gamma$ and provide an embeddingof the subalgebra $\mathscr{A}'_q$ into the quantized algebra of regularfunctions on the corresponding cluster variety.

$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties