It would probably maintain it’s solar orbit and there would be some kind of action with respect to the change in the orbital center of gravity between the moon and the debris field.
If a significant fraction of the earths mass did not achieve a local gravitational escape velocity it would probably re-coalesce into a planetary body.
The “impactor” would have to have some pretty odd physical characteristics to actually punch all the way through with that (visual) effect, like being a Ceres size (~600 mile diameter) mass composed of neutronium (probably ~.6 mile diameter or less, not doing any math here) and travelling at a whole number percentage of C (say 1-2K miles/sec).
It seems theoretically possible that if the whole body broke up, notwithstanding the profuse amount of gasses and water vapor that would be exsanguinated from the interior, there would be a period where objects and human structural components could exist and contain life pending the re-coalescence (hours? days?)... Geology would be a *itch afterwards, give or take a million years.
Just for fun I’ll see your 600 mile slug of Neutronium and raise you my 4,000 mile wad of molten iron the Earth’s core is supposed to be made of. That hit is going to cause a lot of jiggling. You’re thinking the next time I drill a deep well I’d have to strain the Netronies out of the water? If the Neutronium lands on somebody else’s house?