Is there any matter in an electric current?

There is matter flowing. An electric current is a movement of electrons, which are tiny negatively-charged particles. They're very light compared to the nuclei of atoms: about 1/2000'th of the mass of a proton. But they still do have mass, they are matter and they're moving around.

What happens in electric current is, essentially, you've got a conductor where these electrons can move quite easily. You shove some electrons in - so extra electrons enter at one end of the conductor - using a battery or a generator.

Electrons repel one another so that extra density causes all the other electrons to move away a bit.

This produces a sort of "wave"; you can think of it as a wave of movement, going all the way through to the other end of the conductor.

This wave of movement moves at very close at the speed of light - maybe 0.8 of the speed of light - but the actual electrons themselves are only moving at millimetres per second.

So, the actual movement of electrons is very slow, but there is a movement.

The actual signal moves very quickly because it's a bit like a Newton's cradle, where you hit a ball on one end, and then all the other balls transfer that impact all the way along, until the one on the other end flies away far quicker than any of the balls themselves are moving.