what kind of wood is that?
If it is oak or another open grained wood, you will need more than just sandpaper, you'll need filler. Try sanding the wood better, then prime with a high solids primer and sand again, sometimes the primer after a coat or 2 will fill small pitting like you have there.
When you spin the wood on the lathe , do it at the fastest speed and use sharp tools, go slow so as not to have huge chunks of tearout.
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