https://www.theguardian.com/uk-news/2018/mar/05/st-mungos-ho...
https://www.homeless.org.uk/connect/blogs/2018/jan/10/migran...
Actually, lets start with defining what "efficient" actually means in this context? That the goverment spends little money? That it is able to do it's duties effectively?
My personal experience just from San Jose is there are a lot of immigrant homeless from southeast Asia, Mexico, and central America, it just depends on what neighborhood one visits.
Also, people who actually interviewed homeless, found something different from your speculation (I am assuming you didn't actually talk to a lot of homeless people), this story is just Los Angeles but there were a lot of other news stories and it's obviously an issue because the federal government made undocumented immigrants not eligible for federal aid in 1996.
https://www.scpr.org/news/2016/07/14/62582/immigrants-a-larg...