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?
An Unhealthy Looking Calamondin Tree.
Certified GKH Gardening Expert
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.