  The higher step Grushin operators $\Delta_{\alpha}$ are a family of
sub-elliptic operators which degenerate on a sub-manifold of
$\mathbb{R}^{n+m}$. This paper establishes Carleman-type inequalities for these
operators. It is achieved by deriving a weighted $L^{p}-L^{q}$ estimate for the
Grushin-harmonic projector. The crucial ingredient in the proof is the addition
formula for Gegenbauer polynomials due to T. Koornwinder and Y. Xu. As a
consequence, we obtain the strong unique continuation property for the
Schr\"odinger operators $-\Delta_{\alpha}+V$ at points of the degeneracy
manifold, where $V$ belongs to certain $ L^{r}_{{\rm loc}}(\mathbb{R}^{n+m})$.
