It is written Pilagshim or Pilagsham?

Towards the end of Chayei Sarah, we read about the sons of the concubines.

ו וְלִבְנֵי הַפִּילַגְשִׁים אֲשֶׁר לְאַבְרָהָם, נָתַן אַבְרָהָם מַתָּנֹת; וַיְשַׁלְּחֵם מֵעַל יִצְחָק בְּנוֹ, בְּעוֹדֶנּוּ חַי, קֵדְמָה, אֶל-אֶרֶץ קֶדֶם.

6 But unto the sons of the concubines, that Abraham had, Abraham gave gifts; and he sent them away from Isaac his son, while he yet lived, eastward, unto the east country.

Rashi writes on this:

concubines: (Gen. Rabbah 61:4) This [the word [פִּילַגְשִׁם] is spelled defectively [missing the letter “yud”], because there was only one concubine. That was Hagar, who was identical with Keturah. [The“yud,” denoting the plural, is absent, hence Rashi understands that the word פִּילַגְשִׁם denotes the singular. In our Torah scrolls, the plene spelling appears.] Wives are those who have a marriage contract, whereas concubines have no marriage contract, as is explained in Sanhedrin (21a) regarding David’s wives and concubines.

הפילגשים: חסר כתיב, שלא היתה אלא פלגש אחת, היא הגר היא קטורה. נשים בכתובה, פילגשים בלא כתובה, כדאמרינן בסנהדרין (כא א) בנשים ופילגשים דדוד:

and so Rashi is making a diyuk on a spelling of a word, where we have a different spelling! Rashi likely did have the spelling he refers to.

Minchas Shai points out this divergence between our Masoretic text and this derasha. In Bereishit Rabba, parasha 61:4, it states:

ושמה קטורה

רב אמר:

זו הגר.

אמר ליה רבי נחמיה: והכתיב: ויוסף?! ש

אמר לו: על פי הדבור נשאה. היך מה דאת אמר: (ישעיה ח) ויוסף ה' דבר אלי עוד.

אמר לו: והכתיב: ושמה קטורה?! ש

אמר לו: שמקוטרת מצות ומעשים טובים.

אמר לו: והכתיב: ולבני הפילגשים אשר לאברהם?! ש

אמר לו: פלגשם כתיב:.

בעודנו חי, אותה שישבה על הבאר, ואמרה לחי העולמים: ראה בעלבוני!ש

And Rashi, on the basis of this midrash, states that there was only one concubine, who was both Hagar and Keturah.

Gur Aryeh (Maharal) writes on this:

הפילגשים חסר כ׳

דלא היה רק פילגש אחד וקשיא דא״כ מה

תהיה המ״ם של רבוי הפילגשים דלא הוי

למכתב פילגשים במ״ם ויראה דכך פירושו

שהיו בני פילגש אחת מן הפילגשים דעלמא

כמו ויקבר בערי גלעד דכתיב ביפתח לפי

הפשט דפירושו כעיר אחת מן ערי גלעד ה׳ינ

ולבני פלגש אחד מן הפילגשים וכן על עיר

בן אתונות שפירושו עיר בן אחת מן האתונות

אף כאן ולבני פילגש אחת מן חפילגשים ולא

קשיא מידי

But this does not seem to be what the midrash, or Rashi is saying -- that there were really multiple pilagshim, but this is referring to one of the many -- and rather seems to me as Maharal injecting himself into the mechanics of the midrash, and thus ending up making his own novel midrash. If there is a "difficulty" in the final mem that serves no purpose, then so be it! In fact, this is not a difficulty at all -- this is a well known Semitic feature called the "enclitic mem", and Chazal no doubt knew about its existence when forming this midrash.

And see Yefei Toar's commentary on the midrash rabba, where he cites Mizrachi, who understands the shin of pilagsham to serve double-duty, and thus pilegesh sham. See inside. And this, too, need not be if it was intended as an enclitic mem.

Now, Chizkuni (26-27) asks on this that we have it spelled malei, with a yud present. It is so in the sefarim meduyakim! Such that these (plural) are Hagar and Keturah.

And Michlol Yofi writes that Chazal's derasha is known, but we find it malei in all the sefarim meduyakim.

And Radak also says that they darshened it this way because they found it chaser, but we have it otherwise in all the sefarim meduyakim. And so the the Rama writes that all nusachs which are precise have it with two yuds; but that the derasha demonstrates that Chazal (who said this derasha) had otherwise, with a yud missing before the mem.

And so do the Meiri and the author of Shemen Sasson agree; and the masorah that there are two pilagshim which are malei yud twice, one here and one in Esther, demonstrates this.

The question then becomes what we do when an explicit derasha which assumes one spelling contradicts what we have as an established masoretic text. This aspect of it is not for here.

(See, though, the teshuvat Harashba in Beis Yosef's citation, which Minchas Shai cited, which references this difference.)