First, it's important to note that Truman as not asked "to give the go ahead." He was told what was being planned — but he was never part of the chain of command that initiated the strike. (He could have inserted himself into that chain of command, but he didn't. Which is to say, if he had felt the need to, he could have intervened, but he didn't do that — his approach, as Groves later put it, was essentially one of noninterference.)

The place where his awareness of the targets even came up was when Secretary of War Henry Stimson wanted his approval to take Kyoto permanently off of the target list, over the objections of the military. This is the only time that Truman was involved in the question of what kind of target the bomb was going to be used against, and it was Stimson's framing of Hiroshima-as-military-target, Kyoto-as-civilian-target, that is all that Truman knew about the targets until he got damage assessments on August 8th.

So this is the germ of my argument: that Stimson's framing confused Truman as to the nature of Hiroshima. I don't think Stimson intended for Truman to be confused. But I think that was the consequence: Stimson wanted to emphasize that Kyoto was a "more civilian" target that Hiroshima, but in the process he made Hiroshima sound like a "purely military target" (Truman's pre-August 8th words), as opposed to a city with a military base in it.

Anyway, all of this was published early last year in a peer-reviewed volume from Princeton University Press: "The Kyoto Misconception: What Truman Knew, and Didn't Know, About Hiroshima." It is an historical interpretation of evidence, based largely on a very detailed reading of primary sources from the time period, including what little Truman wrote about the bombings prior to August 8th. It also looks at how, on August 8th, his language about the bombings changed very radically — an indication, I argue, that he now understood Hiroshima differently.

As for getting in trouble — Truman never indicated in any way that things were not as he later understood them (Hiroshima as a city). However like many things about bombings this seems to have been about him feeling that ultimately he was responsible (not a bad impulse for a President to have, as opposed to one who blames others for things constantly...), and that ultimately the bombings were worth it. So he justified them, and even made up a totally ahistorical narrative that made him much more central to the "decision to use the bomb" than was actually the case. The place to look for Truman's ambivalence towards Hiroshima is not what he later wrote or said about Hiroshima, but in his later attitudes towards the atomic bomb, which are surprisingly averse for someone who supposedly thought using them on cities was fine.

Anyway — you're welcome to read the article. It's all laid out there, along with caveats, qualifications, places of uncertainty, places of potential counter-evidence, and so on. I am not in any way claiming this is the only possible interpretation. But I find it a fairly persuasive one. But I would... I'm the author of it! Feel free to make up your own mind. :-) I try, in the article, to be as balanced about it as one can be while still pushing one's own thesis, and try to give you as much counter-evidence as I've found for thinking the contrary (ignoring things like postwar memoirs which are very misleading), so when I say "make up your own mind" I'm not setting you up to necessarily agree with me (I try to give you enough information so that you can actually do that).

You may find interesting u/restricteddata's further expansion on just how out of the loop Truman was, especially in the comments below. This specific comment links to their article expanding further on Truman's understanding of the facts on the ground.