In my shul the chazzan occasionally mistakenly skips words in the recital of the day Kiddush of Shabbos, he gets offended at my corrections. Am I obligated to correct him if I catch him mistakenly skipping a word? Sources please.
No, there is no need to correct him in this.
The basic Kiddush for the day on Shabbos is only the berachah: “Borei Peri Ha-Gafen” (Shulchan Aruch, Orach Chaim 289:1).
Beyond the berachah itself, the verses we recite are only as an “introduction,” and they don’t involve any obligation.
Therefore, you don’t have to correct the Chazan – though of course if reciting verse, one should recite them properly.