This problem is related to rotating reference frames. $p$ is a precessing frame or turning table that rotates at $\mathbf{\omega_p}$, and $m$ is a container that spins on the top of the turning table at $\mathbf{\omega}_0$
Using the following expression,
$\frac{\partial \mathbf{r}}{\partial t}|_{i} = \frac{\partial \mathbf{r}}{\partial t}|_{r} + \mathbf{\omega}_r \times \mathbf{r}$
where $\mathbf{r}$ is the position vector, $i$ stands for inertial reference frame and $r$ stands for rotating reference frame.
$\mathbf{u}_i = \mathbf{u}_r + \mathbf{\omega}_r \times \mathbf{r}$
In the container frame,
$\mathbf{u}_i = \mathbf{u}_m + (\mathbf{\omega}_0+\mathbf{\omega}_p) \times \mathbf{r}$,
in the precession frame,
$\mathbf{u}_i = \mathbf{u}_p + \mathbf{\omega}_p \times \mathbf{r}$,
If we move from the mantle (container) to the precession frame (turning table),
$\mathbf{u}_m = \mathbf{u}_p + (\omega_p \mathbf{k}_p - \mathbf{\omega}_0 - \omega_p \mathbf{k}_p(t)) \times \mathbf{r}$
where $\mathbf{k}_p$ is constant and $\mathbf{k}_p(t)$ is constant in magnitude but rotating. In the inertial frame both are the same, but are they supposed to cancel out?
