Firstly, for a bit of the background, I'm a PhD student in engineering at a decent school here in the US. I took most of the basic Physics courses during my undergrad, except for Statistical Physics. My current research has nothing to do with theoretical Physics whatsoever but from last summer, I found myself getting more and more interested in the subject, more specifically in QFT and particle physics. I spent the summer reviewing QM (on the level of MIT's 8.04 and 8.05) and basic math and was able to take two QFT courses at my university in the last two semesters. The two QFT classes covered the standard QFT curriculum up to the renormalization group and non-Abelian gauge theory (basically the first 28 chapters of Schwartz's). I did decently well in these classes, among the top of the class, not that it means anything other than to say that I believe I have the capability to learn the subject.
Unfortunately, I'm in the last year of my program now and currently having to visit a research facility off-campus to finish my thesis. I won't have the opportunity to take any more courses. My goal is to be able to at least read and understand the frontier research in field theory. Now, pretty much on my own, I'm not quite sure how to proceed next. I only have a few hours a day for this so I'm trying to figure out how to best spend my time.
I believe I have a pretty good grasp of the materials I've learnt so far but I'm fully aware that I have a few, quite significant, gaps and holes in my knowledge:
On the Math side, prior to the two QFT courses, I'd only taken the very basics like calculus and linear algebra. I've had to pick up pretty much all the pre-requisites, like group and representation theory, along the way on a need-based basis. While I did understand these topics in the context as utilized in the two QFT courses, I don't feel like I have a strong understanding of the math background.
On the Physics side, I haven't learned statistical physics and general relativity. I also feel I lack a strong understanding of the advanced QM methods, classical field theory, etc.
So the question I'm asking is do I circle back and filling these gaps first or should I force ahead and filling them gradually along the way? My original plan after having finished the two QFT courses was to spend time reviewing Schwartz's book more carefully while learning group and representation theory in a more systematic manner and then perhaps proceed to Weinberg's first two volumes. But after a few chapters, it didn't feel very efficient. With most of the materials already familiar, I found myself skimming through the pages and often missed the subtleties which defeated the whole purpose.
Obviously I have a long way to go and I will have to learn all these topics eventually. What I'm asking is how to spend my time efficiently. How much effort should I put in learning "pure" math. Should I proceed systematically, like starting from analysis and abstract algebra, or should I just keep picking up the requisites along the way? Any advice is very much appreciated.