I also want to chip in with some support forsaoirse’s interpretation of this phrase. I’ve heard it used many, many times, always as an insult. It is used sarcastically, in the sense of: ‘your (cheap, ignorant, duplicitous, lazy) behavior is not the (generous, wise, honest, hardworking) behavior that you think it is’. Or ‘you think you are setting a good example, but you are not.’ However, I have always understood it to have originated from a racist belief that whites are inherently honest, generous, wise, etc., or at least that they should act that way as an example to the ‘less advanced races.’
The phrase may have started out as a genuine complement, used unironically between whites who were feeling self-congratulatory. It could also clearly be used sarcastically by a racist, in the sense of ‘you act more like a (insert epithet here) than a white man.’ The sarcastic usage has become predominant, in the sense of ‘you think that you are being noble, but really you are an asshole.’ Many, perhaps most, people who use the phrase today may be ignorant of its origins and underlying racist connotations, but that does not mean that they do not exist.
For example, I grew up using “gyp” in the sense of being cheated or defrauded. I did not know until my late teens that the phrase originated as a disparagment of the morals of Gypsy peoples. Ever since learning this, I have tried to avoid using the phrase, even though most people who use it are unaware of the underlying slur. Perhaps it can be said that the phrase is now so commonly used in a benign way that its use can be condoned, but I still feel uncomfortable using it and will occasionally point out its history to a person that uses it.
So, it may be that most people that use ‘mighty white’ are not aware of its origins; perhaps they do their own internal folk etymology on it and assume that ‘white’=‘full-spectrum light; pure; noble’ and feel fully justified in using it. But once you have been told of its origins, and that (at least originally) ‘white’=‘member of the white race’, why would you persist in using the phrase? The odds that I would inadvertantly say “what a gyp” to an actual Romany person and offend them is very low, but I still avoid using the phrase, and I don’t think that that makes me hypersensitive or PC.