Does the earth have a nucleus

While the Earth does not have a nucleus like a cell, it does have a core. A nucleus is responsible for protein synthesis, cell division, growth and differentiation. While the Earth's core is responsible for the Earth's magnetic field and tectonic plates. So in summary, the Earth does not have a nucleus.

If you mean in literal terms i. e nucleus as the center then yes Earth has a center.

But if you're asking the nucleus as in the positively charged mass in atom then no.