Ah, more newbie question for y'all. I have greatly appreciated the help
so far.

So far in my experiments, I frequently do this:

1) Turn something rough to round between centers using a spur and live
center
2) Cut a tenon on the head end using a parting tool
3) Remove spur center and replace with Talon
4) Take blank and put it in the Talon, pressing the tenon shoulder
against the jaws
5) Mount Drill chuck (with a drill) in tailstock
6) Drill a hole in blank

In (5) I would expect the "hole" from the live center in the blank to
line up quite close to the drill bit tip, assuming everything runs true.
It doesn't; the drill bit is always off center. The blank runs
*reasonably* true in this scenerio, but not perfectly. The bit is always
off center in the same direction it seems. Of course, with small bits,
this results in the bit flexing toward the little hole the live center
made and gives you a slightly oversized hole. It is a cheap lathe (a
Craftex from busy bee tools, something like Grizzly in the US).

Where am I/lathe/combo going wrong, and what would I do to get
everything lined up better?

PK

PS: The lathe has a rotating head; this is a cause of trouble, as even
tightened down strongly it seems to be able to be moved by a good bump,
i.e. catch. Could this be a cause of trouble? How would one fix it,
outside of a welder?