Siman 140
Presented here for study, not as a practical ruling. The Shulchan Aruch rules by Sephardi custom, the Rema's glosses by Ashkenazi custom, and the Kitzur follows Ashkenazi custom. For a practical question, ask your rabbi.
א הַקּוֹרֵא בַּתּוֹרָה וְנִשְׁתַּתֵּק, הָעוֹמֵד יַתְחִיל מִמָּקוֹם שֶׁהִתְחִיל הָרִאשׁוֹן, וִיבָרֵךְ בִּתְחִלָּה וּבַסוֹף; וּלְהָרַמְבָּ"ם לֹא יְבָרֵךְ בִּתְחִלָּה. הַגָּה: וַאֲפִלּוּ בַּזְּמַן הַזֶּה שֶׁשְּׁלִיחַ צִבּוּר קוֹרֵא, דִּינָא הָכֵי (הר"י פֶּרֶק אֵין עוֹמְדִין).
1 The Laws Regarding An Interruption in the Torah Blessing. Containing 3 S'ifim: One who was reading from the Torah and became silent [i.e. became unable to continue], the one who stands as a replacement should start from the place where the first one began, and should bless before and after [reading]. And according to the Rambam, should not make the blessing before. Rema: And even nowadays where the prayer leader reads, the law is the same. (The R"i - chapter "Ein Omdin" [B'rakhot, Chapter 5)
ב הָעוֹמֵד לִקְרוֹת בַּתּוֹרָה וּבֵרַךְ בְּרָכָה שֶׁלְּפָנֶיהָ, וְקָרָא מִקְצָת פְּסוּקִים וְדִבֵּר דִּבְרֵי תּוֹרָה אוֹ דִּבְרֵי חֹל, לָא הָוֵי הֶפְסֵק וְאֵינוֹ צָרִיךְ לַחֲזֹר וּלְבָרֵךְ.
2 One who stood to read from the Torah, and blessed before it, and read some of the verses, and then stopped and spoke [either] words of Torah or mundane words, this is not [considered] an interruption, and [the reader] does not need to go back and bless [the blessing before once again and start over].
ג וְהָעוֹלֶה לִקְרוֹת בַּתּוֹרָה וְהֶרְאוּ לוֹ מָקוֹם שֶׁצָּרִיךְ לִקְרוֹת, וּבֵרַךְ עַל הַתּוֹרָה וְהִתְחִיל לִקְרוֹת; אוֹ לֹא הִתְחִיל, וְהִזְכִּירוּהוּ שֶׁפָּרָשָׁה אַחֶרֶת צָרִיךְ לִקְרוֹת וְגָלַל הַסֵפֶר תּוֹרָה לַמָּקוֹם שֶׁצָּרִיךְ לִקְרוֹת בּוֹ, יֵשׁ אוֹמְרִים שֶׁאֵינוֹ צָרִיךְ לַחֲזֹר וּלְבָרֵךְ, וְיֵשׁ אוֹמְרִים שֶׁצָּרִיךְ.
3 One who went up to read from the Torah, and they showed [the person] the place where [that person] should read from, and [the person] blessed over the Torah and started to read or [even] did not start to read, and then they reminded [the person] that [the person] is supposed to read a different parasha [section of the reading], and [then the person] rolled the Torah scroll to the place [the person] is supposed to read: there are those who say that [the person] does not need to go back and bless [again], and there are those who say there is a need [to do so].