Both my Variegated Eureka and Eureka lemon is demonstrating the symptom of branch die back, fruit drop, and burnt mark at the petiole. Soothy looking marks appears on the branches too. What is afflicting my trees.

From the one single picture you post, I don't see anything wrong. The particular leaf shown in the picture looks a little under fertilized, but that is not a great concern at present .. Citrus trees more or less always have some twig die back, that is one method the tree uses to maintain its balance between the tree's foliage and root system. - Millet