אורח חיים ק״מOrach Chayim 140

Se’if 1א׳

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).

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. Gloss: And even nowadays where the prayer leader reads, the law is the same. (The R"i - chapter "Ein Omdin" [B'rakhot, Chapter 5)

Se’if 2ב׳

הָעוֹמֵד לִקְרוֹת בַּתּוֹרָה וּבֵרַךְ בְּרָכָה שֶׁלְּפָנֶיהָ, וְקָרָא מִקְצָת פְּסוּקִים וְדִבֵּר דִּבְרֵי תּוֹרָה אוֹ דִּבְרֵי חֹל, לָא הָוֵי הֶפְסֵק וְאֵינוֹ צָרִיךְ לַחֲזֹר וּלְבָרֵךְ.

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].

Se’if 3ג׳

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.

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].