What’s happening to my finger lime?

It’s in Melbourne. Full sun. In a far larger than necessary terracotta pot. It was potted in mostly citrus potting mix and some native potting mix. Has had worm castings spread on top of the soil since then. The pot is well draining. It’s a grafted plant, a few years old, this is the first time it’s produced fruit. It has been healthy looking until last week, there is some kind of dew like stuff on it, branches turning black, and since taking these photos it has dropped most of the emerging fruit. Best I can figure from looking online is…

“Sooty mould, a consequence of honeydew-excreting insects, can be prevented by keeping aphids, soft scales, and mealybugs away from citrus trees”

…but I can’t find any bugs on it. Should I persist spraying it with eco-neem? Try something else? Should I prune off the black branches?