Photos.

That is sooty mold - a fungal growth that develops on the honeydew secreted by sucking insects like aphids, scale, or whiteflies. It is harmless and washes off any fruit easily. Also can be washed of the foliage but that takes a bit more effort :-)

From what is showing, the tree does not look in good shape.....leaves are off color and there's a lot of bare twigs. You want to check closely and treat for whatever insect pests are plaguing the tree and accounting for the development of the sooty mold.

Sooty mold on citrus.

I can almost persuade myself I can see scale in your picture. Check the stems bottom left.

I have had sooty mold on leaves in the past and just wiped it off with my hands when the leaves were wet. For the fruit, I would just wash it well with soap and water and wipe it dry with a paper towel or whatever and then cut open.