  In this paper, we apply the theory of cluster algebras to study minimal
affinizations for the quantum affine algebra of type $F_4$. We show that the
$q$-characters of a large family of minimal affinizations of type $F_4$ satisfy
a system of equations. Moreover, a minimal affinization in this system
corresponds to some cluster variable in some cluster algebra $\mathscr{A}$. For
the other minimal affinizations of type $F_4$ which are not in this system, we
give some conjectural equations which contains these minimal affinizations.
Furthermore, we introduce the concept of dominant monomial graphs to study the
equations satisfied by $q$-characters of modules of quantum affine algebras.
