The Beracha of Me'ein Shalosh, which one recites after eating grain products, wine, or fruits from the seven species, is considered a more "important" Beracha than Borei Nefashot, which one recites after eating other foods. This is true in two senses. Firstly, according to some authorities the recitation of Me'ein Shalosh is required on the level of Torah obligation, while all views agree that the requirement of Borei Nefashot was enacted by the Sages. Secondly, Me'ein Shalosh is more specific than Borei Nefashot; in Me'ein Shalosh, one specifies precisely the category of food he ate – grain products, fruits, or wine – whereas Borei Nefashot makes reference to food generally. In fact, the Gemara speaks of Borei Nefashot with the expression, "Mevarech Ve'lo Kelum," literally, "reciting no Beracha at all." Borei Nefashot is thus considered a "lower-level" Beracha.
Intuitively, then, we might have concluded that when one must recite both a Me'ein Shalosh and a Borei Nefashot, such as if one ate a Ke'zayit of both a Mezonot food (a food made from grain) and apples, he should first recite Me'ein Shalosh and then Borei Nefashot. Since Me'ein Shalosh is considered a more important Beracha, it should seemingly take precedence over Borei Nefashot.
However, Chacham Ovadia Yosef, in his "Yabia Omer," writes that in such a case, when one partook of both Mezonot and apples (or other fruit), he should first recite Borei Nefashot, followed by Me'ein Shalosh. He rules this way because the Semak (Sefer Mitzvot Katan, a work by Rabbi Yitzchak of Courbeil, France, 13th century) rules that the expression mentioned in the text of Me'ein Shalosh, "Ve'al Tenuvat Ha'sadeh" ("and for the produce of the field") includes all produce. According to the Semak, then, the recitation of Me'ein Shalosh covers all foods grown from the ground. Hence, once one recites Me'ein Shalosh over the Mezonot food, he can no longer recite Borei Nefashot over the apples, because the apples were already included in the Me'ein Shalosh. Chacham Ovadia Yosef therefore rules that one should first recite Borei Nefashot, followed by Me'ein Shalosh. Moreover, if a person in this case recited Me'ein Shalosh first, he does not then recite Borei Nefashot, since the apples may have been covered by his recitation of Me'ein Shalosh.
Although Rabbi Moshe Halevi, in his work Birkat Hashem, writes a lengthy essay demonstrating that Me'ein Shalosh always takes precedence over Borei Nefashot, Chacham Ovadia Yosef holds that in this case one should first recite Borei Nefashot.
Thus, if a person ate a Ke'zayit of a grain product, which requires a Me'ein Shalosh, and a Ke'zayit of apples (or other fruit), which requires Borei Nefashot, he should first recite Borei Nefashot, followed by Me'ein Shalosh. And if he recited Me'ein Shalosh first, he does not then recite Borei Nefashot.
