what is the name of the semi-molten rock that forms the mantle?

The semi-molten rock material associated with Earth’s mantle is called magma.

What magma is

• Magma is molten or semi-molten natural rock material found beneath Earth’s surface, from which igneous rocks form.

• It typically forms in the lower crust and upper mantle where high temperature and pressure cause partial melting of solid rocks.

Mantle and semi-molten rock

• Earth’s mantle itself is mostly solid, but parts of it partially melt to produce magma in pockets or chambers.

• This magma can rise toward the surface, and when it erupts and flows outside, it is then called lava instead of magma.

So, the name you are looking for is magma.