I have put off answering your question, actually, because I find it so very ironic (and almost funny). The idea of being "without money" is the exact DEFINITION of "poverty," and therefore early on most everything in this book could be an example of your question. That being said, you're right, remembering the nuances of Jews Without Money, the question has very significant value. Let's clarify first that all the characters here are Jewish immigrants from the Lower East Side of Manhattan at the turn of the century. Here is my best attempt at describing how poverty affects these characters and their immigrant and cultural hardships.
Let's look at Mike's father first. Mike's father was a painter of houses in the region. As happened with many painters of the time, he was eventually injured severely in a fall and couldn't work anymore. Further, as is not surprising due to the quality of paint at the time, Mike's father is further damaged by lead poisoning. Poverty ensues immediately. Mike is disgusted at one point when he finds his dad trying to sell rotten fruit in order to make money for the family. Mike's father, then, is directly affected by poverty.
Mike's younger sister, Esther, actually loses her life due to the family's poverty (and an unfortunate accident). Esther is sent out to gather wood for their wood stove. Due to the poverty of the family, she isn't sent into a forest, but into the streets. Having to pick up scraps of wood on the streets puts Esther in immediate danger. Esther is run over by a truck as she gathers the wood. Esther's death is a further blow to the family.
The family's poverty also affects decisions made BECAUSE OF Esther's death. Herman wants to sign a paper (given to them by a lawyer) to get money from Adam's Express because of Esther's death. Katie, as a traditional Jewish wife, wants nothing to do with this. This would be getting money directly because of the death of their daughter. This is unacceptable in the Jewish tradition. It is considered "blood money" and is the same reason why the Sanhedrin would not accept Judas' return of his thirty pieces of silver. This Jewish tradition is simply that old. It is poverty that guides Herman's decision. Further, indirectly, it is poverty that causes both Herman and Katie to fight.
Another time poverty affects the family is when a United Charities worker visits the home. At this point, the family is starving. They need money. They need food. However, the man questions the way Herman treats Katie. Despite the need (and echoing Katie's earlier traditional decision), Herman throws the United Charities worker out. The family, then, continues in poverty and starvation.
Finally, let's look at Mike himself. There are many ways he is affected by the poverty of his family, but let's look at one striking decision: to leave school at the age of 12 in order to work for the family—even worse, work for the family in disgusting conditions that hardly contribute at all. The end of the book is very telling in regard to Mike's feelings about poverty.
O workers’ Revolution, you brought hope to me, a lonely, suicidal boy. You are the true Messiah. You will destroy the East Side when you come, and build there a garden for the human spirit.
O Revolution, that forced me to think, to struggle and to live.
O great Beginning!
In conclusion, the issues of poverty can be completely separated from the issues of antisemitism that are also present in the book. However, if you are looking for yet another essay topic, poverty combined with antisemitism might be another interesting subject to discuss!