Frobenius nonclassical components of curves with separated variables

We establish a relation between minimal value set polynomials defined over $\mathbb{F}_q$ and certain $q$-Frobenius nonclassical curves. The connection leads to a characterization of the curves of type $g(y)=f(x)$, whose irreducible components are $q$-Frobenius nonclassical. An immediate consequence will be the realization of rich sources of new $q$-Frobenius nonclassical curves.