We follow the custom to recite the Mishna of "Rabbi Hananya Ben Akashya" just before the recitation of Kaddish that precedes Baruch She'amar in the morning. The reason for this practice is that sometimes, the prayer service begins before a Minyan has arrived, and the tenth man comes in right after La'menase'ah, before the Kaddish that precedes Baruch She'amar. In order to allow the recitation of Kaddish, a Minyan must have been present for the reading of words of Torah. We therefore recite "Rabbi Hananya Ben Akashya" to allow the recitation of Kaddish if the tenth men arrived right at that point, before Kaddish.
If fewer than ten men are present in the synagogue when it is time to begin Minha, the congregation may begin reciting the sections of the Tamid and the Ketoret, but they should not begin Ashreh before the tenth man arrives. According to some opinions, the half-Kaddish following Ashreh can be recited only if a Minyan was present for Ashreh, and so the congregation should wait for a Minyan to arrive before beginning Ashreh. However, if they recited Ashreh without a Minyan, and the tenth man then arrived, then, according to some Poskim, Kaddish may nevertheless be recited, because our custom is for the Hazzan to recite two verses – "Tikon Tefilati Lefanecha" (Tehillim 141:2) and "Hakshiba Le'kol Shav'i" (Tehillim 5:3) – just before the half-Kaddish preceding the Amida at Minha. The Kaf Ha'haim (Rav Yaakob Haim Sofer, Baghdad-Jerusalem, 1870-1939) writes that the custom in his time was to recite Ashreh while waiting for the tenth man, and to then rely on the recitation of these two verses before Kaddish once the tenth man arrives. However, the Mishna Berura ruled that at least three verses must be read to allow the recitation of Kaddish. Moreover, the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Od Yosef Hai (Parashat Vayakhel), indicates that an entire chapter of Tehillim should be recited with a Minyan before Kaddish. Therefore, it is preferable to wait for a Minyan before reciting Ashreh, though if Ashreh was recited without a Minyan, and the tenth man arrived after Ashreh, the congregation may rely on the two verses of "Tikon Tefilati" and "Hakshiba."
If the entire morning Pesukeh De'zimra service was recited without a Minyan, and the tenth man arrived after Yishtabah at the conclusion of Pesukeh De'zimra, the Hazzan may recite at that point the half-Kaddish preceding Barechu. Likewise, if, during Arbit, the tenth man arrived only after the reading of Shema and all its blessings, the Hazzan may recite the half-Kaddish before the Amida.