The black spots are caused by apple scab. This although undesirable shouldn't be the primary the apples are small. There are quite a few reasons for small fruit. Do you have any more information on the tree and/or a picture of it.
Generally valid and good advice, but I can't let the suggestion to remove all mulch andthe top 3 to 5 inches of soil to prevent scab, go unchallenged. It is true that the Venturia fungus overwinters in the leaf litter from the apple trees. (Leaves from other species do not harbour apple scab, and can be left as mulch.) These fungi sporulate when spring rains splash the fungi. That is, it is the surface fungi that are at play. Actually, allowing the understory to grow up reduces the burden of contamination because it traps the spores in the undergrowth so they do not get up onto either fruit or leaves. Removing the topsoil from the bases of the trees will have no beneficial effect, and considerable adverse effects.
More practical ways of managing scab are 1) spraying sulphur. (I don't) 2) growing scab-resistant apples, (I do, some 50 odd varieties, mostly heritage varieties, but some recent cultivars bred for scab resistance. I would highly recommend Williams Pride, out of the PRI breeding program, for those who want a bright red shiny apple with good flavour - totally scab-free.)
And I would second the comment from the French gentleman - if your tree is a wild seedling, it will make wild seedling apples, which are in the vast majority of cases, small and scabby - and sour. But if that is what you have, the thing to do is top-work it to a desirable variety. And best to choose a scab-resistant one while you are at it.
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