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21. Mincha:

Mincha is recited prior to Tashlich. See next!

Avinu Malkeinu: The Chabad custom is to sing the Niggun of Avinu Malkeinu prior to the start of Mincha.[1] This custom was initiated by the Rebbe.[2] 

Saying Tzidkascha Tzedek by Mincha of Shabbos R”H:[3] When [the first day of] Rosh Hashanah falls on Shabbos we are accustomed to omit the recitation of Tzidkascha from the Mincha prayer of Shabbos.

Mincha Erev Shabbos:[4] When the 2nd day of R”H coincides with Erev Shabbos, Avinu Malkeinu is omitted by Mincha.[5]

Veani Sefilasi:[6] When Rosh Hashanah coincides with Shabbos the verse of Veani Sefilasi is recited by Mincha of Shabbos as is usually done.

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[1] Hisvadyos 1984 1 p. 26-27 “The reason for the custom to sing the Niggun of Ainu Malkeinu of the Alter Rebbe prior to each prayer on Rosh Hashanah is in order to mention the merit of our forefathers, the merit of the Alter Rebbe.”

[2] Otzer Minhagei Chabad p. 58

[3] 598:1

Background:

Although when a regular Yom Tov falls on Shabbos we omit Tzidkascha by Mincha, nevertheless on R”H that falls on Shabbos we say Tzidkascha being that it is a day of judgment and it is proper to say “Tziduk Hadin” on the day of judgment, and so is the custom of Sefardic Jewry. [Tur; Beis Yosef; Michaber 598:1] However there are opinions [Rama 597:1; Ran; Minhagim] that rule that even on R”H that falls on Shabbos one is to omit Tzidkascha being the verse states “Mishpatecha Tehom Raba” and one is to avoid arousing judgment on the day of judgment. Furthermore, this day is Rosh Chodesh and one does not say Tzidkascha on Rosh Chodesh as explained in chapter 292. Practically the custom is like this opinion in these provinces. [Admur ibid] Practically even amongst Sefardic Jewry some communities are now accustomed to omit it, and each community is to follow their custom. [Kaf Hachaim 598:1]

[4] 584:5; 602:2; Meil Tzedaka; Rivash 512; M”A 584:3; M”E 584:11; M”B 584:4; Shaareiy Tziyon 584:7; P”M 584 A”A 6; Kaf Hachaim 584:7

Other opinions: From the Levush it is implied that Avinu Malkeinu is recited on Mincha Erev Shabbos. [Kaf Hachaim ibid]

[5] The reason: From Admur ibid it is implied that this is because it is forbidden to make personal requests on Shabbos. Vetzaruch Iyun as it is not yet Shabbos and hence should not carry this prohibition? [See Kaf Hachaim 584:7]

[6] Sefer Haminhagim p. 133 [English]; Levush 598; Mateh Efraim 598:3; See Otzer Minhagei Chabad p. 140