https://www.urban.org/urban-wire/two-american-experiences-racial-divide-poverty
There are distinctions in poverty
https://www.washingtonpost.com/news/wonk/wp/2015/08/12/black-poverty-differs-from-white-poverty/?utm_term=.fd6c9535268d
Even in poverty, there are privileges and differences if you want to do a side by side. So, in summation, you are wrong. Poor people can be privileged based on race on other factors.
“Privilege” includes numerous societal aspects and factors. It’s not money or class alone. And, it may be best seen depending on the reference point.