Your lemon tree has an insect that is secreting the sticky stuff. It is probably either scale, whiteflies or mealybugs. One solution that should work on any of these is horticultural oil. Follow label instructions carefully regarding how strong to mix the product and make sure the spray mix is kept well mixed by shaking periodically. Oil kills scale by coating and suffocating the insect, not by poisoning them. Therefore getting thorough coverage is critical if you are to get effective control. Reapplications a week or two later may be needed to control any scale missed with the first application.
The lemon will drop any fruit that is too much for it mature. Don't thin the tree yourself. Just keep it in as much light as you can and keep the soil moist to prevent drought stresses.
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