In addition, if anyone has answers:
Wikipedia sources this claim to The Superman Files, by Matthew K. Manning. It also says that a comic strip in 1945 where Lex blasts Superman with particles from a cyclotron was pulled, but that implies it was after publication, so the title question isn't relevant in that case (though the follow-up question is.)
I haven't heard of a 1944 censorship case; all of the reputable literature is about April 1945. The way voluntary censorship worked in the latter case is that they could prevent syndication, reducing the impact of how far it would spread. This was done by the Office of Censorship, which had in 1944 promulgated regulations about mentioning atom smashers, atomic energy, nuclear fission, etc. in American news media.
There were cases of publishers being pressured into not publishing prior to publication, but as you note that could only happen in cases where for some reason the fact of the impending publication became known to the Manhattan Project security people. They did spent an inordinate amount of time running down "rumors" and "leaks" of this sort, no matter how trivial (and most were trivial). They spent far more time on this than, for example, searching for spies — it made up the bulk of their activities. But they were not psychic.
As for why censor it — the goal was to try and prevent as much press discussion and speculation about nuclear physics as possible, so that any foreign readers would not get curious. Because you could "see" the Manhattan Project — even as a blank spot, much less in a million tiny leaks — if you got curious about these things. It was a somewhat quixotic philosophy, to be sure, and unique to the wartime period of voluntary censorship and "absolute secrecy," in which the fact that there was a secret was in fact one of the secrets. This was an enormously difficult thing to maintain and there were major errors and disclosures.
(The Germans, it is interesting to note, had no real knowledge of the Manhattan Project. But they did notice some of the popular writing on atomic energy and did wonder if that indicated official US interest. But they never followed up on it in a serious way, for a variety of reasons. The Japanese were similarly in the dark. The Soviets, on the other hand, both saw the absence of publication as a sign that a major effort was under way, and they also ended up with several highly-placed spies.)
Chapter 2 of my recently published book on the history of nuclear secrecy in the US talks at length about the voluntary censorship program as well as the attempts to ferret out leaks and rumors. One of the interesting things I found is that the rate of mentioning "banned topics" (like cyclotrons, uranium, etc.) did not really change as a result of the official censorship order. Rather, there was something that decreased discussion of these topics, but it came earlier: as more and more of the US scientists were drawn into the war effort, there were far fewer sources for these kinds of stories. So it is not clear that these censorship efforts really mattered all that much.
For a detailed discussion of the Office of Censorship and the 1945 Superman case, see Patrick Washburn, "The Office of Censorship's Attempt to Control Press Coverage of the Atomic Bomb During World War II," Journalism Monographs 120 (1990), 1-43; Superman is on 20-23.
I have requested a copy of The Superman Files; I'll admit I am a little skeptical on the face of it! Update: I got the Superman Files book from the library, and the page given on Wikipedia doesn't even contain this claim. I'll flip through it to see if I actually can find this claim but I'm pretty skeptical. The book is an "in-universe" take on Superman, it is not a work of history in the slightest.
The source that wikipedia cites says simply that "(How the Department of Defense knew about the storyline is unclear)" [1] - worth noting they use the anachronistic name for the institution here as evidence they may not have dug particularly deeply. A bit of further exploration turns up the story appearing on the Superman home page in slightly different terms [2], suggesting it may be a story which has "done the rounds" for some time.
As implied by your question there are some obvious problems with this story; "government agents" did not possess the surveillance capabilities to be able to intervene in the production of a publication _during the creation_. Censorship during the war was coordinated by the Office of Censorship and worked according to self-censorship following a voluntary code primarily because the Office didn't have the capability to monitor and censor in the manner described in the story. The FCC monitored ratio broadcasts, and the Office of Censorship monitored press (possibly including comic books), but only had the capability to respond either reactively or in response to complaints or people voluntarily coming forward to request advise [3].
Two examples of failed censorship with respect to the Manhattan project help to illustrate this: 1) In 1944 The Cleveland Press published an article about a "Forbidden City" research facility out in New Mexico, identifying it as being run by Robert Oppenheimer and describing the security measures at the base [4]. 2) Also in 1944 a MBS show called "Confidentially Yours" broadcast that the military was working on splitting the atom and would soon have a nuclear weapon [5]. Though the Office of Censorship reacted to these swiftly and treated them seriously it only had the ability to respond reactively e.g. demanding all recordings be destroyed.
One thing you'd definitely care to read is a column on the subject from the New York Review of Science Fiction [6]. I won't summarise the whole thing here but suffice it to say the Office of Censorship put out a circulate in 1943 to a wide swath of publications requesting no publication of any articles or broadcasts on the subject of atomic weapons. My best guess is that the story is made up, but if you are determined for it to be true I think the most likely explanation is that someone involved in the editorial process had seen this notice, or had it passed on to them, and they hit pause on the publication; and the part about shadowy government agents paying a mysterious visit has likely been added to the story later as embelishment.
As to the other question about other cases, most of the sources here also make mention of 'Crime Paradise', a similar piece from 1946 about the test at Bikini Atoll, though as [6] points out this would have been after wartime censorship ended and the Office of Censorship shut down so there's even less reason to suppose it was true and claims that it was written earlier seem flatly false given the clear reference to a 1946 event. The same piece mentions an instance of the arrest of Philip Wylie by Army Intelligence to prevent the publication of a novella about atomic weapons, though there it would appear his published was the cause having written to the War Department to seek permission.
Bibliography
[1] Mason & Kantor (2013) - Superheroes; Capes, Cowls and the Creation of a Comic Book Culture.
[2] https://www.supermanhomepage.com/comics/comics.php?topic=articles/supes-war
[3] Sweeney, MS (2001) Secrets of Victory, the Office of Censorship and the American Press and Radio in World War II. p59
[4] http://blog.nuclearsecrecy.com/2013/09/20/worst-manhattan-project-leaks/
[5] Sweeney, MS (2001) Secrets of Victory, the Office of Censorship and the American Press and Radio in World War II. p201
Wait. I thought the actual bombing was the first time most people even heard about nuclear tech. Was the ability to make a nuclear bomb common knowledge before we dropped them?