An unhealthy looking Calamondin tree.
The bottom part of the plant has dead branches and very unhealthy looking dead leaves but the top has new leaves growing and it has beared fruit. Can I cut this tree down a little and start it fresh again?
Branch dieback can be a sign of disease or environmental problems, or it can be normal as the tree grows. Please see these articles for more information:
You can prune out the dead branches, but be sure to sterilize your tools with a 10% bleach solution after making the cuts to prevent spreading any possible disease organisms.
If there are a lot of dead branches, check this article for diseases that match your tree's symptoms, especially in the "aboveground dieback" section.