This OU article by Rabbi Zivotofsky summarises many different opinions:
Two extreme positions are cited by the Tur (OC 142): Either there is no need to correct any error and doing so will embarrass the baal korei, or, as the Shulchan Aruch (OC 142:1) and Rambam (Hilchot Tefillah 12:6) rule, even minor grammatical errors must be corrected. In this vein, the Sha’arei Efraim (3:16) advises chastising the baal korei even for minor errors. The Ramah (OC 142:1 and see commentaries) states a compromise position that requires correcting only mistakes that alter the meaning of the word.
Adding a name is obviously not a smaller error regarding a vowel, and for such cases:
The commonly accepted practice follows a third opinion, that of the Baal Hatanya, which requires the baal korei to re-read from the beginning of the phrase in which the error occurred. This ruling applies even if the phrase contains God’s name.*
* See Rabbi Hershel Schachter, “Lesser-Known Laws of Torah Reading,” Journal of Jewish Music 7:1-11, paragraph 46 where he also cites the Tosafot Anshei Shem to Mishnayot Brachot 2:3.
Phrase in this context is a meaningful unit of a verse as indicated by the teamim. However, there are others, who require the whole verse to be repeated, while some rabbanim are more lenient, and only require to correct the word and continue from there.
The repetition raises the concern of saying the divine name in vain, and it is discussed by the Tzitz Eliezer in connection with those, who finish first the verse before correcting. An argument for this practice can be found in Megillah 22a:
כל פסוקא דלא פסקיה משה אנן לא פסקינן ליה
Any verse that Moses did not divide, we may not divide. (Sefaria)
Despite the gemara, he cites R. Eliezer Papo, who argued in the Chesed laAlafim that this is problematic, if there is a divine name later on in the verse, so one should not finish it in such cases. Although it is still another debate, whether one should return to the beginning of the verse if the first half contained a divine name.