It may not be a root, but a 'sucker', or secondary 'trunk' coming up, in which case you'd want to snip it as low down as possible to keep it from taking nutrients and strength from the main trunk (a lot of deciduous trees grow suckers). If it is a root, then cover it with a little soil til it hardens and becomes a regular surface root. It may be needed to support new growth.

It is hornbeaM, by the way.

Unless the tree is in really bad soil, I'd let it be for now. If you're pulling on the tree and it moves, it's a sign that there is only a little feeder root down there. Don't disturb the tree too much though by constantly moving it around in the pot. That will discourage new roots.

I'm with Mark on this one--why are you pulling on the tree??? If it moves when you're pulling on it, it means that your "friend of a friend" didn't tie the tree in well at the last re-pot. Every time you pull on the tree, you're potentially damaging the small feeder roots that sustain the tree--slowly killing it. If you want to keep doing bonsai, PLEASE be gentler with your trees!!!