homework and exercises - Gravitational field of sphere containing a spherical cavity

I'm just struggling a little with this question:

A uniform sphere, of radius $R$, contains a spherical cavity of radius $\frac14R$, whose centre is $\frac38R$ from the surface. The diameter passing through the centres of the sphere and cavity meets the surface at points $X$ and $Y$. Find the ratio of the gravitational field at $X$ and $Y$.

My attempt at the solution goes something like this:

Using the superposition principle, the gravitation field due to the whole mass is equal to the sum of the gravitational fields due to the remaining mass and the removed mass.

The gravitational field due to a uniform solid sphere is zero at its centre. Therefore, the gravitational field due to the removed mass is zero at its centre.

The gravitational field due to the solid sphere is equal to the gravitational field due to the remaining mass. Now we know g acts towards the centre of the sphere. As such, both the gravitational field of the combination of the sphere and removed mass and the gravitational field of the sphere only act in the same direction, so we can use the scalar form of the equation.

Therefore the gravitational field is given by $g=\frac{GMr}{R^2}$.

Then insert $r=-R$ and $R$ for the gravitational field at $X$ and $Y$.

But this doesn't seem to be correct as it is just the same as if the removed mass wasn't there..... Have I gone wrong in my logic somewhere?