Its not clear to me that that that's true; there is complicated intersectionality at play. Poor whites are more likely to have social networks and other advantages that provide things useful to getting out of poverty, and may -- due to racism and the racially unequal distribution of wealth and power -- have better responses from people in power independent of their own social networks.
OTOH, they may also be more likely to have social networks that are less understanding of an accepting of poverty.
Whether this is a net advantage or net disadvantage varies, among poor whites, considerably from individual to individual.
Group privilege exists mostly in terms of average conditions across groups (and may not thus be particularly useful in discussing the positions of individuals.)
Taking White privilege as a given (which I think does accurately reflect reality, in terms of on-average advantage), I'm not sure that poor White privilege (compared to, poor non-White ) is valid; its certainly not a necessary corollary of White privilege.