In this article, we prove null-controllability results for the heat equation associated to fractional Baouendi-Grushin operators $$ \partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u=\un_\Omega h $$ where $V$ is a potential that satisfies some power growth conditions and the set $\Omega$ is thick in some sense. This extends previously known results for potentials $V(x)=|x|^{2k}$. To do so, we study Zhu-Zhuge's spectral inequality for Schrödinger operators with power growth potentials, and give a precised quantitative form of it.