What philosophers and in what writings, if any, have attempted to explain or defend Descartes's rationalism in respect to the "cogito ergo sum" fallacy pointed out by philosophers like Russell, and Wittgenstein?
I believe you're referring to the fallacy of inferring the “sum” (or “I am”) part from the “cogito” (or “I think”) part, right? The “ergo” (or “therefore”) makes it sound like Descartes is expressing an argument which has as its premise that he thinks, and the conclusion that he exists.
The potential fallacy in this representation of the Cogito statement is that a more substantive meaning of “I” is usually read into the conclusion than is read into the premise. The support for the premise is just the introspective observation of the occurrence of thought. The conclusion, however, can be understood to mean the he, Descartes, exists—a guy named René who's French and has long hair and wrote letters to princesses, and everything else that's part of being René Descartes. That stuff is not entailed by the premise about the mere occurrence of thought, hence the accusation of fallacy.
The consensus response to this in recent Descartes scholarship is that the Cogito is not an argument at all, where either part is inferred from or logically-entailed by the other. Rather, the thought that Descartes thinks is indubitable is something like “there's thought and there's existence,” where the two parts of that claim are bound up with one other and both are perceived simultaneously.
Indeed, that is how Descartes generally expresses the point, outside of the mention of the Cogito sentence. In the second Meditation, he writes:
So, after considering everything very thoroughly, I must finally conclude that this proposition, I am, I exist is necessarily true whenever it is put forward by me or conceived in my mind.
There's no inference there, and therefore no fallacy. Instead, there's just a thought that whenever thought is a necessary truth.
Now of course, moving beyond that statement to prove any other statements is of course THE big cartesian problem, and one that Descartes is not thought to have satisfactorily solved. But this version of the cogito, which Descartes is thought to have held, at least avoids the possible fallacy you mention as potentially intrinsic to the very expression of the sentence.
Centuries before Descartes made up that (maybe perceived) fallacy, Avicenna the persian muslim philosopher had already explained why it is a logical fallacy to argue from an issue of the self (in this case thinking) for the existence of the self. Because in any such argument the existence of the self is already presumed as it is impossible to experience any issue of the self (such as thinking) without having first experienced the self itself. The glaring fallacy is manifest even by looking at the apparent semantics of the English translation of the argument that starts with a presumed "I" in the premise ("I think, ") to prove the "I" in the conclusion ("therefore I am.").
PS: Even if the famous statement is not meant to be a logical proof (as ChristopherE and Asphir Dom suggested in comments), but a reference to a human transcendental experience (which is supposed to be beyond proof and self-evident); it still holds true that the first object of transcendental experience is not any mental/psychological process (e.g. thoughts, feelings etc) associated with human self but rather the very direct experience of the self which precedes them all. Nowhere this is better explained than in the theory of "knowledge by presence" which was first proposed by Al-Farabi and later refined by Avicenna. In knowledge by presence, it is argued, the subject and object of knowledge are united -- either because of the identity of the two or one being the intimate part of the other --, therefore the subject can directly experience the object of knowledge without any medium. Such a knowledge is thus self-evident, infallible and needless of proof. Examples are human experience of one's very self, one's thoughts and feelings.
The theory is very significant as it bridges the gap between philosophy and mysticism, as in the latter most statements of truth are based on direct experience rather than logical proof. This allows objects of knowledge by presence to serve as self-evident premises for philosophy. That's how muslim philosophers for the first time reconciled Aristotelian philosophy with Platonic.
I fundamentally agree with @Christopher - the cogito argument is an argument, but not an inference.
There is no inference of the form : "if p, then q".
On all the complex issue, it is worth reading in SEP the entry on Descartes' Epistemology and in The Cambridge Companion to Descartes, ed.John Cottingham (1992), at least : Louis Loeb, “The Cartesian Circle” and Peter Markie, “The Cogito and Its Importance”.
About Russell's critic, see in SEP entry :
Among the critics, Bertrand Russell objects that “the word ‘I’ is really illegitimate”; that Descartes should have, instead, stated “his ultimate premiss in the form ‘there are thoughts’.” Russell adds that “the word ‘I’ is grammatically convenient, but does not describe a datum.” [B.Russell, A History of Western Philosophy (1945),page 567] Accordingly, “there is pain” and “I am in pain” have different contents, and Descartes is entitled only to the former.