Influential Theory in the Economics of Poverty As far as I can tell, research on the economics of poverty is very empirically driven, and this is probably appropriate. What kind of economic theory has been developed though? I'd also be interested in theory of some influence too. To avoid wholly opinion-based answers, let's measure influence by how the theory directed other research or anti-poverty measures. As an example, I know papers have been written on the theory of poverty traps, but I'm not sure that concept is a contribution from theorists so much as some theorists developed a model for what might have already been an empirical observation. Are there any cases where concepts started in theory?

Links to papers are good, but the question is also about history to some extent, so it would be nice if you can show a trail of this paper influenced that paper or this policy.
To extend @Majoko's comment, you may be very interested in the book Poor Economics which discusses many of the issues you note. It specifically discusses issues with theory, and of course has a lot of empirical work to back it up.

Perhaps on the other end of the spectrum is general equilibrium theory applied to poverty and the developing world. you should really look into Robert Townsend's work. His "Thai Project" is a massive dive into some of the questions you're asking. A reasonable place to start is probably with his books. 

As an aside, I was originally hooked on his work by his fantastic application of microeconomic general equilibrium theory in The Medieval Village Economy, which has sadly gone out of print (although I see there are now some used copies finally selling for less than $70, which is nice). He takes very sparse data and asks what theory might be able to tell us -- then very carefully and purposefully walks the reader from a most basic model through many extensions to explore important possible variations of the medieval experience. It's really an excellent example of applied general equilibrium work, and how theory might still be tested/explored with little data.

This isn't my area of research, but I'm very interested to see other answers appear here.