It depends on how big the pit is. Do I assume correctly that you sanded or cut into it; that it didn't appear on the surface of the wood. If it's small, I would consider leaving it alone, especially if you are sure that it doesn't open up into a larger void. I'm afraid that trying to cover these things up it is like putting makeup over a pimple. Yes, you may not see the pimple anymore but you have this little off color/off texture spot on your face that says "zit under here". If it is a real problem, like it does open up into a void, remove the handles and start all over. This is just one of the very few problems that can come from using wood, even stabilized wood. All materials have their advantages and disadvantages. It is also more likely to happen with burl or spalted wood.
Doug