The black stuff could be black sooty mold, but would like to see an image to confirm. Burford holly can be a host to different kinds of scale insects, which contribute the excretion (called honey dew) that the black mold grows on.

I've also seen very heavy growths of BSM on plants that were growing underneath the canopy of a larger plant infested with scale or aphids.

Anyway, I'd like to see a picture of these black spots as well as the black stuff on the leaves. Look along the stems as well as the leaves for little bumps...those would be the scale insects.