Kaddish is recited at several points in the morning Shaharit service, marking the transition that we make from one spiritual realm to another. The first section of the morning prayer service is the Korbanot, which deals with the "Olam Ha’asiya" – the world of action, as it speaks of the performance of sacrifices in the Bet Ha’mikdash. Kaddish is therefore recited after the section of Korbanot, before Pesukeh De’zimra, which represents the "Olam Ha’yesira." The next Kaddish is recited after Yishtabah, which marks the conclusion of Pesukeh De’zimra, before we proceed to the section of "Yoser Or," which signifies the next realm, the "Olam Ha’beri’a." In theory, another Kaddish should be recited after "Go’el Yisrael," before the Amida, as the Amida brings us to the realm of "Asilut." However, the Sages did not want any interruption made between the theme of Ge’ula (redemption) and the Amida prayer, and so no Kaddish is recited at that point.
The custom in some Minyanim in our community is not to recite the Kaddish after Korbanot until after Hodu, before Baruch She’amar. This custom stems from the practice of the Kabbalists in Yeshivat Bet-El. The custom there was that if a Minyan was not yet present at the conclusion of Korbanot, but was present after Hodu, the Kaddish would be recited at that point, right before Baruch She’amar. This custom is recorded by the Kaf Ha’haim (50:6). Apparently, the transition from the "Olam Ha’asiya" to the "Olam Ha’yesira" takes place at Baruch She’amar, and so the Kaddish may still be recited even after Hodu, before Baruch She’amar. This practice, it seems, gave rise to the custom to always recite the Kaddish after Hodu.
Interestingly enough, Hacham Bension Abba Shaul (Israel, 1923-1998), in his Or Le’sion (vol. 2, 45:1; listen to audio recording for precise citation), extended this ruling even further, claiming that the Kaddish may even be recited during Pesukeh De’zimra. Meaning, even if the tenth man did not arrive until some point after Baruch She’amar, the congregation should interrupt the Pesukeh De’zimra recitation, recite the verses, "Baruch Hashem Elokeh Yisrael Min Ha’olam," and then recite Kaddish. The work Dibreh Shalom (listen to audio recording for precise citation) questions whether Hacham Bension actually issued this ruling or whether this was written in error by his students, because it seems clear that once the congregation begins reciting "Baruch She’amar," the transition has been made and Kaddish may no longer be recited. (The Dibreh Shalom adds that the transition may be made even without Kaddish, just as a person who prays without a Minyan is able to make this transition from one realm to the next without Kaddish.) Indeed, the accepted practice does not follow Hacham Bension’s ruling, and the Kaddish after Korbanot cannot be recited after Baruch She’amar.
Summary: It is customary in some congregations for the Kaddish following the Korbanot section to be recited only after Hodu, just before Baruch She’amar. If there is no Minyan at that point, and the tenth man arrives after Baruch She’amar, the Kaddish may not be recited.