Siman 598

For mincha they say "Ashrei," "U-va Le-Tziyon," and seven blessings as in maariv and shacharit. And if it falls on Shabbat, they read from the weekly parsha. The shaliach tzibbur repeats the prayer and kedusha, as on a weekday. And only on Yom Kippur do they also say the kedusha of musaf at mincha, but not on Rosh Hashanah. And there are some who even on Yom Kippur do not say it, but only as on a weekday. And they say "Avinu Malkeinu," and if it falls on Shabbat they do not say it. And know that the custom of our city, when they pray mincha on Rosh Hashanah immediately after musaf—they do not say "Avinu Malkeinu" even on a weekday, because they said it during shacharit.

If it falls on Shabbat, the Tur wrote in the name of Rav Amram that we recite "Tzidkatkha Tzedek," for even though on every Yom Tov we do not recite it, nevertheless on Rosh Hashanah, which is the Day of Judgment, it should be recited. And some say not to recite it, for in any event it is a Yom Tov, and also Rosh Chodesh. And such is our common custom. And if it is because of the Day of Judgment—the judgments have already been sweetened after the Musaf prayer.