When writing the paper about completed experiments, should you cite work you hadn't read before?

An important realization I had while writing the first few papers of my career is that a paper/thesis/any kind of publication should not be a recollection of your path through this research. You will have made many experiments as part of the research that need not actually appear in the paper because they're not important for what you're trying to show.

Instead, a publication is a way for you to explain insights you have gained during this research. In which order you came to these insights, in which you read other papers, or in which you did experiments is -- generally -- not important for the telling of the story.

So, regarding the original question whether one should cite a paper one hadn't read before the research started: it can't be answered with yes/no. Whether a paper should be cited (as asked in the title) has nothing to do with when one read the paper. It has everything to do with whether you think it's a useful reference for readers to have. In the body of the question, there is a variant: whether it's "legitimate", and there the answer clearly is "yes". In particular, we add references to papers all the time that were pointed out by reviewers -- long after the research was concluded for the most part.

Should you cite work you hadn't read before doing the research?

Yes.

You are ethically obligated to cite work that helped you conduct your research. That in no way implies you should not cite work for other reasons.

You should cite every paper that might help your reader. In my opinion, it is also acceptable to cite a relevant paper just because it is traditional to cite it, or because you liked the paper.

I strongly believe that one has an obligation to cite "prior art", whether or not one was ignorant of it or not. If one is ignorant of things, they do not directly influence one's work, but to write/publish a paper with no mention of other peoples' work is irresponsible, possibly to the point of being immoral.

Yes, I have heard a few very-well-known mathematicians say, as a joke but not really, that they avoid reading their competitors' papers so they "won't have to cite them".

And, indeed, some people seem to be in some sort of solipsistic mode, in that they feel no obligation of scholarship or helpfulness to other people. Once I myself caught on to the (obvious, really) deficits of this approach, I did indeed start trying to be more conscientious.

(Plus, interestingly, especially in mathematics, an amazing number of ideas were rediscovered over-and-over... in the last 250 years, at least. The supposed "lack of modern " turns out not to mean that people didn't have the ideas!)

So: cite what you used, cite what you looked at and/but didn't use, and cite what you belatedly discovered. I know other peoples' views are different.