When reviewing a manuscript for the second time, is it fair to comment on aspects I did not realise the first time?
I believe that it is entirely reasonable to make major requests of a manuscript on second review, as long as it is maintaining your basic standard for acceptance rather than moving the goalposts.
For example, I have had the experience where things about a paper were very unclear to me in the initial review because of shortcomings in the authors' presentation. Upon revision, those things became much clearer---but not in a good way. The extra information shed light on serious flaws in the authors' work, which has led me to in some cases recommend further major revision and in other cases recommend rejection.
On the other hand, some reviewers seem to like asking authors to do entirely new work, not because it is necessary for publication, but because the reviewer wants to know the answer or thinks it will make "a more interesting paper." This happens most frequently with "glamour" journals. I believe this sort of "moving the goalposts" on publication is not appropriate even in the first review, and doubly so in a revision review.
You certainly shouldn't refrain from suggesting improvements or corrections simply because it is the second round of refereeing. Go ahead and include them, as it can do no harm. It's also fine to point out minor things like typos or awkward sentences that you didn't catch the first time. And if you have just now discovered that the paper's main result is wrong -- or likely to be wrong -- you have an obligation to communicate that fact.
On the other hand, I would usually refrain at this stage from insisting on major new changes to the manuscript that I hadn't requested previously; especially things like
- A major reorganization or rewriting
- New experiments or analysis
If there have been new developments in the field that significantly affect the status of results in the paper, then it might be appropriate to insist on substantial rewriting or additional research, but otherwise I would say it was really your job to catch these issues the first time around.
If there are things genuinely wrong with the paper, you should definitely ask from them to be improved. If the issues are with things that could probably be handled better, yet which are not fatal errors, you might want to go easier on them. However, even in the latter case, there is no reason not to send back the paper for additional minor revisions before recommending acceptance.
When I review a paper with a lot of problems, yet which I feel could probably be published with major changes, I try to include an explicit statement in my first review that looks something like this: "At a minimum, the authors need to make the changes that I have suggested in order for this paper to be publishable. However, given the major problems that I have identified, it is not possible at this stage for me to judge definitively whether the paper's conclusions will be justified once the changes are made; it is possible that a heavily revised paper will still turn out not to be satisfactory." This makes it clear that I cannot adequately judge the correctness of the paper without the major changes being made. In my experience, such papers usually are publishable after the major changes are made (although often with another round of minor revisions), but it a significant number of cases, the revisions only serve to make clear that there are fundamental problems with the paper, and I ultimately have to recommend rejection.