Patents, products and improvement patents are separate concepts. The vast majority of products include inventions that were never patented or for which the patents expired. Anyone who wants to can make those products without any permission. Patents limit inventions for a limited time and anyone making a product containing a patented invention during the patent term would need a license from the patent owners.
Improvements on patented inventions can be patented, provided they meet the criteria of being "new and non-obvious", among other administrative requirements (enabling disclosure of best method, etc). Whether or not the underlying invention is patented is not particularly relevant to the improvement patent. You could (in theory) patent an improvement on something originally invented 100 years ago.
However, owning an improvement patent on something that IS still under patent for the underlying invention does not give the owner of the improvement patent any legal right to make, use, sell or import any product with the underlying invention without the necessary license for that invention as well.
Having said all that, you must be careful to understand that a patent is a property right that includes specific and broad claims to the invention. If you do ANYTHING that copies the invention, or anything EQUIVALENT, then that could be a patent infringement.
Under your example, say I patent a wooden pencil cylinder containing a graphite extrusion. It would be an obvious equivalent to use "something like wood" instead of wood, or something like graphite to substitute in my own pencil. These would be patent infringements. People get sued for infringement. If I were, on the other hand, to patent "a method for efficiently sharpening a wooden pencil", that invention would not necessarily infringe the pencil patent, since it does not actually require a wooden pencil as part of the invention itself; the pencil is supplied externally. What if I were to patent a copy machine and its ink cartridges so that no other cartridge would fit and the cartridges could not be refilled? Would a new invention that bypasses the patented invention in the cartridges to enable others to refill the cartridges be an improvement or an infringement? Even the federal judges don't know yet.