We investigate a gauged Higgs-Yukawa model within the framework of the Functional Renormalization Group. In previous studies, it has been shown that the inclusion of higher dimensional operators beyond $\phi^4$ in the bare potential of the Higgs field $\phi$ can diminish the lower Higgs-mass bound and thus shift the onset of new physics towards higher scales. In a subsequent step, we included generalized Yukawa interactions, e.g. $\phi^3 \bar{\psi}\psi$, which have the potential to lower the Higgs mass analogously to the case of generalized Higgs potentials. Indeed, we also observed a decrease in the lower mass bound, however, the mechanism at work is similar to the one present in the case of generalized Higgs potentials. In particular, the effects of both types of higher dimensional operators do not stack.