In this study, the $(h$- $\phi {)}_R$ and $(h$- $\phi {)}_M$-contractions with two metrics endowed with a directed graph are examined using auxiliary functions. We propose a set of criteria that guarantees the existence of common fixed points for our contractions. This leads to a generalization of previous results in the literature. Towards our accomplishments, we establish affirmative results that demonstrate solutions to a class of nonlinear two-term fractional differential equations with nonlocal boundary conditions. To further corroborate our major findings, we also provide instances.