Calculate next point on sphere a certain distance away

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I have a sphere where I know the radius, origin and a single point $p^1$ on the surface by its cartesian coordinates ('world' x/y/z).

I'm struggling trying to calculate the position of another point $p^2$ on the surface a certain x/y spherical distance away from $p^1$.

Any pointers in the right direction or solutions would be much appreciated :)

Image explaining what I'm trying to accomplish