Here are a bunch of variations of the sentence.
I gave him a paper to read.
I gave him some paper to read.
I gave him the paper to read.
I gave him the papers to read.
I gave him paper to read.
I gave him some papers to read.
I gave him papers to read.
These are all correct English sentences that have different meanings. Some of them are ... strange.
In (1), "a paper" refers to some document (academic or otherwise). Its specific identity is not the central point of the sentence, the act of giving is more central.
In (2), "some paper" refers to pieces of paper. It is strange, because when paper has writing on it, usually the writing is the important part; "some paper" has the connotation that the important part is the actual sheets. I'd expect nearby sentences to talk about how the paper would be written on (printers maybe?)
In (3), "the paper" could either refer to a newspaper, or some specific document. If a newspaper it would probably be a physical object; if a specific document, it could be on a computer or a file emailed. The document is specific, not general, as in "a paper"; either I'd expect the identity or description of this document from other nearby sentences, or it would be a sentence defining the importance of the document where the central point of the sentence is "the paper".
In (4), we are now talking about more than one document. They again are specific, and are either defined nearby, or this sentence puts them at the forefront and the important piece of information being conveyed in the sentence.
In (5), the meaning is nearly identical to (2). Using it like (3) or (2) feels wrong: contrast this with (7).
In (6), the added plural in paper means we go from sheets of paper, to many documents (or newspapers). This is the plural of (1); the papers are used interchangeably, we aren't talking about specific documents, but rather an undifferentiated sample of some population of documents.
(7) is a variation on (6) or (4). The papers feel less "anonymous" than (6), but less specific than (4). Contrast this with (5). The plural here prevents it from being about sheets of paper, forcing it to be about documents. The lack of some/the makes it both less specific and less non-specific at the same time.
In some regional dialects, passing the newspaper to someone would be "I gave him the papers to read", as a newspaper is a bunch of papers. In other dialects, that would be strange, but recognizable.