Many of the earlier works cite the Talmud Yerushalmi that states:  One who recites Parshas Hamon every day is assured that his food will not be lacking.

The source of this Yerushalmi however is unclear. Sefer haManhig written by R' Avraham ben Nathan is one of the earliest works to mention this custom and states it is found at the end of Yerushalmi Tractate Yoma. The Tashbetz, a student of the Maharam of Rothenburg cites this custom as well. However commentators on the side of his work note they have not found this Yerushalmi inside either. 

One of the great and early Chasidic masters, Rav Menachem Mendel M’Rimonov, who was one of the earliest and strongest proponents of the chasidic movement throughout Poland and Galicia has become the popularly recognized source of a special segula to recite these passages on Tuesday the week preceding the shabbos on which we read Parshas B’shalach. 

Interestingly, the Shu”t Migdal Na’eh writes that if one looks at the writings of R’ Shlomo Yehudah Weinberger, which is where this segulah from the Rimonover is first found, what he actually says is that on Tuesday Parshas Beshalach, the Stropkover Rebbe told him that the Rimonover recommended saying Parshas Hamon every day.

The custom is to read the actual words of the Torah twice, and the text of the Targum Onkelos once to fulfill this segulah. For the full text of Parshas Hamon click here.