This subject gets center stage every year. The question is not whether there is a segula in reciting Parshas Hamon. There is ample reference to this. The issue is whether there is any significance whatsoever to Tuesday of Parshas Beshalach.
I have yet to see a single reputable source for this. I am acquainted with a rather big Talmid Chochom, a Rosh Yeshiva in one of the Chassidishe Yeshivos, who stated publicly that he remembers exactly when this “minhag” was formed. It was approximately in 1947, and was part of a casual conversation among a group of baalei batim in a shul on the Lower East Side.
Meanwhile, virtually any reference to Tuesday Beshalach postdates that. I would be open to any factual challenge to this.
BTW, I also know a devoutly Chassidishe family that comes from Rimanov, and are also descendants of Reb Mendele ZT”L, and they know nothing of any such mesora from Reb Mendele about this “minhag” of Tuesday Beshalach. Again, I am open to any factual challenge.
Meanwhile, I believe there is no such minhag, and that is was fabricated. It is attributed to Reb Mendele, who was an advocate to recite Parshas Hamon every day. The originator had the mazel, and his fabrication went viral.