Power Fractal noise functions are in World Space by default. This means the noise function is anchored to the absolute coordinate positions of the overall scene, rather than to coordinates/position relative to the object's position. So if you move the object, the area of the noise function that influences it changes too. This is what the "Translate textures with planet" option in the Planet object is for. So you could replace the sphere with a Planet and use that option.
The other approach would be to add a Transform Input shader in-between your Power Fractal and the displaced object, then copy the new position of the object to it (Translate and Rotate functions). This will give you the noise function as it appeared if the object was at 0,0,0. If you need to replicate a look for an object that was elsewhere you'll have to work out the difference in coordinates between the old and new position of the object to put that into the Transform shader.
- Oshyan