  In this paper, we study the minimal affinizations over the quantum affine
algebras of type $C_n$ by using the theory of cluster algebras. We show that
the $q$-characters of a large family of minimal affinizations of type $C_n$
satisfy some systems of equations. These equations correspond to mutation
equations of some cluster algebras. Furthermore, we show that the minimal
affinizations in these equations correspond to cluster variables in these
cluster algebras.
