When I called, he came a running.
My first inclination was that it's an article and the participle is being used as a gerund, but that doesn't make sense structurally.
My second is I wonder if it comes from Latin or a Latin-based language influence. For example, European Portuguese puts an "a" between:
He came running. (He came a running)
Ele(He) veio(came) a(a) correr(running).
This suggested duplicate question is not in fact duplicate. It doesn't provide a grammatical basis. It doesn't explain what it is. Is a being used as an article or a preposition or something else? It is (couched midway in the third of seventeen paragraphs of one of three answers on the alleged duplicate post) postulated to be a preposition. But this has no basis in grammar because the only prepositional definition of a is per, as in each, because a does not mean on, not in any dictionary from OED to Merriam-Webster to American Heritage or to any other I could find. This other post asks for the story behind a; I'm asking what tenet of grammar justifies using a, which this other post does not illuminate.