What are the different reasons given for the minchat chavitin, minchat chinuch of the cohein hedyot, and the minchat chinuch of the cohein gadol. Are there any significant pieces written by the acharonim on these topics?
The Sefer ha'Chinuch (Mitzvah 136) discusses the Minchas Chavitin of the Kohen Gadol. He explains that as the Kohen Gadol is the messenger of Bnei Yisrael to Hash-m, it is befitting for him to have a constant Korban like the Korban Tamid (twice a day). The fact that he brings such a Korban everyday focuses him on his uniquely important role in life. For an interesting explanation why specifically the Minchas Chavitin is brought daily, see the Hemek Davar (Vayikra 6:13, see also Gilyonei Hashas on Bava Basra 164a).
The Minchas Chinuch is an inaugural Minchah appropriate for Kohanim who reach a new stage of Gedulah and Kedushah in bringing Korbanos. Many Acharonim liken the Minchas Chavitin of a Kohen Gadol to a constant Minchas Chinuch (see Hemek Davar and Drash Moshe Vayikra 6:13). See Aznayim l'Torah ibid. for the ideas behind these Menachos.
All the best,
Yaakov Montrose