Answer: If the browing problem is affecting the entire plant rather than just a single branch or two, it is most likely disease; if only a few spots are affected, then it is most likely insect related. Leyland cypress is sometimes host to spider mites and other pests but without knowing for sure which one, it is difficult to make control recommendations. Carefully inspect the affected plant parts, looking for signs of insect activity: webbing, frass (bug poop), active insects, etc. You may have to use a magnifying glass. If you can't find insects, look for signs of diseases such as oozing sap, fungus, cankers, etc. Or, cut off one of the affected branches and take it to your cooperative extension office, Master Gardener clinic, or well staffed nursery for positive identification of the problem. Once you know what you're dealing with, you'll be able to decide which control measure to employ.
Best wishes with your landscape!
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