The Shulhan Aruch (Orah Haim 267) rules that Tahanunim are omitted from the Minha prayer service on Friday afternoon. The Kedusha (sanctity) of Shabbat begins to set in even before the actual onset of Shabbat, and thus it is inappropriate to recite Tahanunim during Minha.
The Peri Megadim (Rav Yosef Teomim 1727-1792) writes that this applies even if a person recites Minha Gedola meaning, early in the afternoon and even if a person prays immediately after Hasot (midday as defined by Halacha). The implication of the Peri Megadim is that Tahanunim becomes inappropriate already from Hasot, and that the critical factor here is not the Minha prayer, but rather the time of day. As such, if somebody wakes up late on Friday, such as if he was ill, and he began praying Shaharit late in the morning, he must be aware that Tahanunim should not be recited after Hasot. If Hasot passes by the time he completes the Amida and reaches the point where Tahanunim are normally recited, he does not recite Tahanunim, and thus instead of "Ana" he recites "Yehi Shem." This Halacha is especially relevant during the winter months, when Hasot is at around 11:45am, and thus it can certainly happen that somebody who is ill completes the Amida at Shaharit after Hasot. In such a case, he omits Tahanunim.
This ruling appears in the work "Az Nidberu."
Summary: Tahanunim should not be recited after Hasot on Friday. Thus, at Minha, or even at Shaharit if one reaches the point of Tahanunim after Hasot, one recites "Yehi Shem" instead of "Ana."
