Each week, during Minha on Shabbat, we recite the Pasuk in Tehillim, "Va’ani Tefilati Lecha Hashem Et Rason Elokim Be’rob Hasdecha Aneni Be’emet Yish’echa" ("My prayer to You, Hashem, at this time of favor – G-d, in Your abundant kindness, answer me with the truth of Your salvation"). The Tur (Rabbenu Yaakob Ben Asher, 1269-1343) explains (in Orah Haim 622) that we recite this verse because David Ha’melech here observes the difference between the Jews and the other nations of his time. When members of other nations would celebrate, David said, they would eat, drink, get drunk and engage in all kinds of illicit behavior. The Jewish people, however, celebrate Shabbat by eating, drinking, and then returning to the synagogue in the afternoon to pray. This verse essentially describes what we are doing at Minhah on Shabbat – returning to the synagogue after feasting – and it is therefore appropriate to recite it as part of the Minha prayer.
According to this reason, it would seem that this verse should not be recited at Minha when Yom Kippur falls on Shabbat. Since we do not eat on Yom Kippur afternoon, this verse is not relevant to the Minha service on that day. However, some Kabbalists maintain that there is also another reason for reciting this Pasuk, explaining that the time of Minha on Shabbat is a special "Et Rason" – an auspicious time for earning G-d’s favor – and we therefore recite the Pasuk that speaks of an "Et Rason." If so, then the afternoon of Yom Kippur, which is a time of even greater "Rason," is certainly an appropriate occasion for reciting this Pasuk.
Practically speaking, however, the accepted Halacha is that this verse is not recited at Minha of Yom Kippur, even when Yom Kippur falls on Shabbat. This is the ruling of the Mahari Semah (17th century), in his work Nagid U’mesaveh, where he writes that even according to the teachings of Kabbalah, this verse is not recited on Yom Kippur. The Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) also codifies this ruling, which he bases on the teachings of the Arizal (Rav Yishak Luria of Safed, 1534-1572). This is, indeed, the ruling of Hacham Ovadia Yosef, in Yalkut Yosef – Yamim Nora’im (p. 402; listen to audio recording for precise citation).
Summary: The verse, "Va’ani Tefilati," which is normally recited at Minha on Shabbat, is not recited during Minha on Yom Kippur, even when Yom Kippur falls on Shabbat.
