In order to answer this, let us first look at what the AGIL acronym stands for. Talcott Parsons argues that every social system needs four characteristics in order for it to survive. First, it has to be able to adapt (this is where the “A” comes from) to and transform its environment. Second, it has to be able to create and achieve goals (the “G”). Third, it must integrate (I) its various parts. This means that it needs to regulate the relationships between various groups within it. Finally, it must be able to renew the patterns within the culture that motivate people. Parsons calls this the “latency” function (L). Each society needs to perform all of these functions. Problems within a society tend to be caused by a breakdown in one or the other of them. Let us examine how this is so using two of the key problem areas that you identify in your question.
Poverty can be seen as a breakdown in the “A” function. In a large society like ours, the A function is carried out by the economy. The economy is the part of our society that transforms our environment and which allows us to adapt to it. When the economy does not function properly, we have poverty. Parsons’ AGIL framework cannot tell us exactly what is wrong with the economy, and therefore it may not be particularly useful with respect to that problem.
The issue of immigration policy could be seen as a problem of the “I” function in the United States (it is also caused by the strength of our economy and the weakness of other economies). The I function allows our society to integrate its various parts. We argue about immigration partly because we are afraid that excessive immigration will give us too many disparate parts (ethnic groups) to integrate well. We also argue about it in part because various parts of our society (people of different political outlooks) do not relate well to one another and do not trust each other.
In these ways, we can use Parsons’ AGIL framework to analyze various issues that arise in our society.