I would probably want to reframe food poverty as simply poverty, and then talk of household food insecurity and/or hunger (although the jury is still out there for the perfect language!).
A brilliant book to read is Hannah Lambie-Mumford's book 'Hungry Britain, the rise of food charity'. And yes, I'd say that poverty and hunger is heavily affected by the socio-economic and institutional environments (the book describes it really well). It's important to remember that systems don't really have boundaries - everything is interconnected, so you can look at hunger through the socio-economic angle, and some of that will inevitably stem from and/or have implications on food systems themselves.
I am not familiar with IFSTAL's 'food systems conceptual framework', but what I would recommend is rather than trying to find where this issue fits into the framework, instead start from the issue and think about what links/relates to it. What are the social, economic, cultural, technological, political drivers that affect the issue? They all influence poverty and hunger somehow. Frameworks are useful to help us think about and conceptualise complexity, rather than to force us to categorise everything in that one particular combination.
As for how this space is evolving at the moment, it is interesting to see:
On a positive note, the huge civic response to help feed the nation, from mutual aid groups to a call for farmers to give 1-acre of their farms to communities for self-sufficiency, business models adapting quickly (eg: Leon, Nandos, Cook Foods) and food networks synchronising efforts. From a food citizenship point of view, there is a huge revival of communities and realisation of our own power as citizens. We are seeing new networks emerging, existing ones strengthening, and with it a whole new power structure that will forever change food systems.
On the more negative side, there are some tensions from already well-established organisations redistributing food to those in need and emerging networks who are less experienced, with concerns that these newcomers are accidentally doing things 'the wrong way'. There is also huuuuge pressures on food redistribution programmes to feed even more people with less resources (staff self-isolating, no adequate storage for the massive influx of surplus food coming from the foodservice industry shutting down...). Finally, there is a risk that we further entrench the belief that emergency food is the only response, rather than think long-term about what resilience looks like, and what systems are needed to be put in place to avoid chaos from future shocks. The RSA's Food Farming and Countryside Commission has just announced a big call-out for data to beging to think about what that long-term resilience could look like (https://twitter.com/FFC_Commission/status/1245660937132158976).
What I would also note is that we have temporarily gone from a complex (food) system, to a chaotic one. Until we have found some sort of new balance, it is harder to apply a systems lens to what is currently happening.
At the Food Ethics Council, we had a Food Talks on Tuesday about some of these questions already. Summary notes + link to the video recording here: https://www.foodethicscouncil.org/foodtalks-emergency-responses-to-covid-19/
Hope this helps,