Is there a word to describe the repeated usage of the same word at the beginning of a phrase or sentence?
For example: "As if he were trying to say something, as if he intended to explain, as if he meant to clarify the point." Is there a word for that literary device?
The post above has identified the term you are most likely looking for - anaphora. In the study of literature and writing, anaphora refers to the first words in successive sentences and phrases.
In rhetoric, an anaphora (Greek: ἀναφορά, "carrying back") is a rhetorical device that consists of repeating a sequence of words at the beginnings of neighboring clauses, thereby lending them emphasis.
Anaphora can be understood as a special form of parallelism also, where a sentence structure or sentence type is repeated. So, parallelism is another (more general) term that applies to the example you've provided.
Anaphora also falls into the rhetorical category of repitition.
I think that it is an anaphora.