There have been instances where we had 9 people in the shul, and we began to daven Mincha Shabbat afternoon. We finished our individual Shmoneh Esreh, when the 10th person appeared. Can we go back and read the Torah, or since we are past that point in the service, we cannot do so?
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The mitzvah of reading from the Torah on Shabbos, YomTov and Mondays and Thursdays takes place during tefillah but is not dependent upon it. Certainly the preferable mode of reading from the Torah is during davening but the mitzvah lasts all day and when a minyan is available they should read from Torah. (Mishne Berurah 135:1, Shaar Hatziyun 3 and Aruch Hashulchan 135:6,7, Dagul Mervavah, Mahari Asad 58. However, some maintain that if mincha time has arrived, the Torah should not be read. See Mikraei Kodesh (Grodzinsky), Klal 13:5 and Shaarei Kedusha 10,11)
Years ago I spent my afternoons in a kosher slaughterhouse and the shochtim started working before sunrise and continued until noon with breaks after their hourly shifts. They were unable to form a minyan when they davened Shachris but, during lunch hour, when all the shochtim were off duty and could form a minyan, they read from the Torah.
I checked with my LOR and he said that if someone makes the minyon after everyone else has davened, he can say borchu, shmoneh esrai with haiche kedusha, and the minyon can lain. The reason for this is that since a minyon is present, he can daven with what must be said in the presence of a minyon even though nobody else is davening. I am deliberately not dealing with duchening (for those whose minhag is to duchen every day) as I do not have the sources available or the time to go into it. My LOR did not discuss that case. The main question is leining if the tenth person comes after everyone else has finished davening (especially Shabbos Mincha when shmoneh esrai follows leining). Duchening does not apply for Shabbos Mincha in any case.
Logic might seem to indicate that if everyone else can stay and no kohen has had kiddush (which would prevent the kohanim from duchening) then the kohanim should be able to go in front of the aron and duchen (during Shacharis). However, this is logic only based on the idea that a kohen who has davened already can duchen at a different minyon. I have not seen this explicitly in the various sources that I have read, but I have not seen anything to refute it either. This is the same logic that allows the Torah to be read at a separate time.