I'm not good with the title for this scenario:
Suppose that there where 10 men in the shul - 9 everyone knew, and the 10th no one knew. He wore a yarmulka, so everyone assumed that he was Jewish.
They read the Torah on a Thursday morning. After davening was over, the rabbi welcomed the unknown man to the shul, and asked him his name. He says, "Avigdor McDonald. I am thinking of converting, so I am exploring various synagogues."
Oops! It turns out that they read the Torah without a valid minyan. Since Torah reading is a mitzvah for the congregation, let's say that the shul finds another Jew - a "real" one - who walked in after davening. Should they repeat the Torah reading?
Let's say, they discover this info about the not-yet convert in the middle of Torah reading. For example, they call him up for Shlishi and that's when he says he's not Jewish. Should they finish the Torah reading, anyway, or what?