Overview of cap865

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career advice_how to address an academic gap and is a second MSc the solution?

I think applying for the MSc makes perfect sense and I don't expect you would have to justify your decision to do it in any kind of defensive way. The reason itself for the academic gap should be a non-issue and the only practical consideration now should be whether - as you say - your knowledge is outdated as a result of your time away. That's exactly what you're looking to address with the MSc, so it won't be seen as superfluous. I would just focus on making sure you can frame it in a positive and progressive way when asked, so yes - don't describe it as though you're trying to repair any perceived 'damage' resulting from the gap or make up for shortcomings in your recent PhD applications. You have real-world experience and awareness that anyone coming straight out of undergrad will lack, and you want to augment that experience by bringing your knowledge in a new field up to the state-of-the-art!

I'd add that it's easy to get disheartened by rejections, but remember that (last I heard) the success rate for PhD funding applications is something like 15%, so any rejection could simply be a matter of bad - or even average - luck, and it's possible you don't need the MSc if learning what you need during the PhD is an option. Still, if you can spare the time and money, I expect the MSc will give both you and and prospective supervisors confidence that you're back in the game and make the eventual PhD more pleasant, since you'll feel in your element rather than under pressure to catch up.

How long to study a (STEM) module that provides important background for your research

I'm interested in knowing how you manage your time when studying fields that provide useful background for your research, but aren't its primary focus. I feel a 'full' understanding isn't strictly necessary if you only need enough context to follow what's going on when elements of the field crop up in your research - at least, I hope so, since academia is vast and time is limited! Still, I suspect that simply attending lectures (or equivalent time spent reading) wouldn't bring sufficient understanding, as you would normally build on the lectures with revision/further reading and be actively solving example problems, in order to be able to understand later content.

As a baseline, let's say you want an understanding of a module for which 150 hours of total study are suggested and whose core content is covered in 30 hours of lectures, sufficient to:
- score 50% in every question on its written exam,
- follow in real time an explanation of the perfect exam solutions,
- spot when a technique/result is used in a research paper and understand well enough to continue reading after a few minutes of thought/referring back to learning materials.

With access to any lecture notes, problem sheets and textbooks you need for self-study, how would you go about gaining this level of understanding, and how long would you expect it to take in total? Does the answer change if you're returning to a module you already studied, but so long ago that you would fail the exam if you took it today?

Thanks in advance!

[Probably-superfluous context: I'm looking to return to academia with a PhD in Mathematics (my undergraduate degree), having done some postgraduate masters in Computer Science and Machine Learning and a few years in industry in Data Science. I'm interested in topics I didn't get to study as an undergraduate (e.g. Formal Logic, Set Theory, Model Theory, Modular Forms) and feel I need to study in some depth before committing to PhDs in that area. On top of these, there's a vast list of relevant modules that I have studied before, but which I feel so rusty in that I would almost need to learn them from scratch - but perhaps it will all come flooding back to me in a fraction of the original time. I'm trying to squeeze all this into the limited amount of time I have to study alongside working full-time.]