= Solution
The centre-of-<mass> coordinates are
$$
a=h\sin\theta\cos\phi,\qquad b=h\sin\theta\sin\phi,\qquad c=z+h\cos\theta.
$$
Differentiation and cancellation of cross terms gives
$$
\dot a^2+\dot b^2+\dot c^2=\dot z^2+h^2(\dot\theta^2+\dot\phi^2\sin^2\theta)-2h\sin\theta\,\dot z\dot\theta.
$$
Insert this into the supplied <kinetic energy> and subtract the gravitational potential $mgc$. Thus
$$
\boxed{L=\frac{A+mh^2}{2}(\dot\theta^2+\dot\phi^2\sin^2\theta)+\frac C2(\dot\psi+\dot\phi\cos\theta)^2+\frac m2(\dot z^2-2h\sin\theta\dot z\dot\theta)-mg(z+h\cos\theta).}
$$
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