If one is saying the Shema, got distracted while reciting the first verse, but then finished the rest of the shema into the time after the zeman, do they have to recite it again because of being distracted while reciting the first verse?
Not concentrating on the first passuk invalidates the reading of Shema. The MB 63:14 discusses it and explains that one has to repeat the entire first parsha if he only realized after he finished:
ואפילו אם כבר קרא כל פרשת שמע והיה אם שמוע ואח"כ נזכר שלא קרא פסוק ראשון בכונה צריך לחזור ולקרות כל הפרשה שמע דאם לא יקרא רק הפסוק ראשון הוי כקורא למפרע דאינו יוצא כדלקמן בסי' ס"ד. ואם נזכר באמצע פרשת והיה אם שמוע יגמור עד ויאמר ויתחיל ויקרא שמע עד והיה אם שמוע ויחזור לפרשת ויאמר:
So in your case, he would have to go back and repeat the first parsha. Since it’s after the zman, he should havein mind like he’s reading the Torah, like anyone who says Shema after the zman
Kitzer Schulchan Aruch Siman 17:9 says
"If one stopped in the middle of [his] recitation of Shema - and his interruption was due to an unavoidable circumstance that forced him to stop - for example, he had to go relieve himself, or excrement was found in the room, - if he waited for a duration of time in which he could have completed the entire Krias Shema, he must go back to the beginning of Krias Shema, [and begin again]."
The footnotes say that "this refers only to circumstances in which he is halachically forbidden to recite the rest of the Shema. However, if he was forced by other circumstances, it is not considered an interruption with regard to Shema, and he need not start again from the beginning (Mishnah Berurah 65:2)."
The footnotes also say "This (duration of time) is measured for each individual, based on the time that it would take him to recite the entire Shema (Rama 65:1). It does not mean the time it would take him to recite from where he paused until the end, but rather the time it would take him to recite the entire Shema from beginning to end (Mishnah Berurah 65:3).
If one was saying the shema, and got distracted, that does not seem like it was a situation where he is halachically forbidden to recite the shema. Therefore you would not need to start again.
If the distraction was indeed halachically forbidden, it is important how long this interruption lasted - depending on how long it takes the given individual to say the entire Shema.