The Noda Be'Yehuda (Tzlach, Brachos 37b, here) suggests that the Matza that we eat with Pesach Sheini does not have all the requirements and restrictions that apply to the Matza we eat on the first night of Pesach. He bases this on the fact that the Gemara requires a drasha that brings over to the Korban Pesach Sheini those idiosyncratic laws that we were taught by the Korban Pesach Rishon. The absence of any such equating-drasha regarding the Matza of Pesach Rishon and that of Pesach Sheini indicates that there is no such equivalence. The Noda Be'Yehuda is very tentative about his chiddush, though.
Reb Chaim on Pesachim 120a (page 33 of Al Hashas) says clearly that there are two mitzvos of Matza on the first night of Pesach: the Matza derived from "בערב תאכלו מצות" which pertains irrespective of having a Korban Pesach, and צלי אש ומצות על מרורים יאכלוהו- which is the same as על מצות ומרורים- the mitzva to eat the Korban Pesach with Matza and Maror. He brings the Rosh (in Arvei Pesachim #25) that says that even when Maror was a mitzva de'oraysa, namely, when there was a korban Pesach, you didn't have to eat a kezayis of Maror. You just had to eat the Korban Pesach with Maror; the Maror was an adjunct to the Korban. How do you eat the Korban? With Maror. And Matza. According to Reb Chaim, then, the separate Mitzva of Matza that is adjunct to the Korban also does not necessarily have to be a kezayis.
Reb Itzaleh in his Zecher Yitzchak, (and as brought in the Mekor Baruch, by Reb Chaim Stein's uncle, who says he heard it from Reb Itzaleh when he was a young boy,) uses this idea, which he said independently, to answer a kashe everyone asks on the Yerushalmi brought in Tosfos in Kiddushin 38a DH Akruv. The gist of what he says is this: The Gemara says that the first year we came to Israel, in the time of Yehoshua, they only ate grain after they brought the Korban that was mattir Chadash. There are many ways to read this Gemara, but Tosfos brings that the Yerushalmi asks, why didn't they eat Matza the first night of Pesach, because it was Chadash? So why don't we say עשה דוחה את לא תעשה, that the mitzva of Matza supercedes the issur of Chadash. He brings some baalebatishe answers from the Yerushalmi. But everyone asks on the Yerushalmi that the Gemara in Pesachim says that you're only yotzei with Matza that is susceptible to having had an issur of Chametz, and so you're not yotzei with Matza of Tevel, because an item that is already assur cannot become doubly assur, אין איסור חל על איסור. The Gemara asks, but Chametz is an issur kollel, it prohibits many other things besides this tevel, so it should be assur as chametz even if it's already assur as tevel. The Gemara answers that it's going to according the Reb Shimon who doesn't care if it's a broader issur, you still say אין איסור חל על איסור. So everyone asks on the Yerushalmi, since our Gemara in Kiddushin is assuming that all the grain in the world was assur when the Jews first came to Israel, than Chametz is not an issur kollel, it's not a broader issur, so even the Rabanan should agree that it was only assur as Chadash, and that the issur of Chametz did not affect it, and so you couldn't be yotzei with such a Matza which couldn't have been assur as Chametz. So Reb Itzaleh answers that the rule that you're only yotzei with Matza-which-could-have-had-an-issur-of-Chametz only applies to the Matza of "בערב תאכלו מצות", and the Yerushalmi is really talking about the Matza of על מצות ומרורים יאכלוהו, the Matza that is adjunct to the Korban Pesach, and none of the side rules of Matza apply to that Matza, including the idea that it has to have been susceptible to the issur of Chametz.
So the evolution of this vort is as follows:
1. The Noda Be'Yehuda said that the side rules of Matza don't apply to the Matza of Pesach Sheini.
2. Reb Chaim says in the Rambam and in the Rosh that both the Matza and the Maror that are eaten with the Pesach, even at Pesach Rishon, are mere adjuncts to the Korban Pesach, and are completely unrelated to the general mitzva of Matza, and you wouldn't need a kezayis of Maror or that Matza to be yotzei the matza-and-maror-as-adjuncts-to-the-korban mitzva.
3. Reb Itzaleh also says that the matza-as-adjunct doesn't have all the dinim of the Matza of Mitzvas Matza; this being the case, we can answer the kashe on the Yerushalmi, because not only don't you need kezayis, you also don't need could-have-been-assur-as-chametz and not-matza-ashira and shmura-le'sheim-mitzva. So the Yerushalmi is asking about using Chadash for צלי אש ומצות על מרורים יאכלוהו, which is basically the same as the Matza of על מצות ומרורים, namely, the matza-with-the-korban. The Yerushalmi is not asking regarding the matza of בערב תאכלו מצות.
The practical application would be that if you're not well, and have a hard time eating regular matza, you still have to eat regular matza for the basic mitzvah because you are only yotzei with לחם עוני. But for Korach, you can go ahead and use Matza Ashira. Or matza that is not Shmura.
By the way, the Rosh that Reb Chaim brings down says that the only reason we need to eat a kezayis of Maror is because the language of the bracha is "al achilas maror," since the word 'Achilas' necessitates a kezayis. As far as the Torah is concerned, you never needed a kezayis of maror. Everyone (beginning with the Shagas Aryeh in siman 100) asks on the Rosh that if Midoraysa you don't need a kezayis, why did Chazal formulate the Bracha with the word 'Achilas' which then mandates eating a kezayis. Reb Moshe in the Igros OC III 66, and other achronim, says that the Rosh never meant that. Reb Moshe says that the Rosh only meant to explain why Karpas doesn't need a kezayis while maror does, and on that the Rosh answers because Karpas has no bracha, and so there's no need for a kezayis; it's just a remez/zeicher, while Maror is a takana gemura to do like it was done when there was a Korban Pesach. On the other hand, Reb Moshe in the last piece in his Dibros on Pesachim does assume like everyone else in the Rosh, and klers whether the matza and maror that accompany the korban are more like the four minim by Sukkos or more like the Lechem by Korban Todah.
I just came across the Sfas Emes in Pesachim 99b, the first piece in AP, that also says this.
By the way, the Rosh that Reb Chaim brings down says that the only reason we need to eat a kezayis of Maror is because the language of the bracha is "al achilas maror," since the word 'Achilas' necessitates a kezayis. As far as the Torah is concerned, you never needed a kezayis of maror. Everyone (beginning with the Shagas Aryeh in siman 100) asks on the Rosh that if Midoraysa you don't need a kezayis, why did Chazal formulate the Bracha with the word 'Achilas' which then mandates eating a kezayis. Reb Moshe in the Igros OC III 66, and other achronim, says that the Rosh never meant that. Reb Moshe says that the Rosh only meant to explain why Karpas doesn't need a kezayis while maror does, and on that the Rosh answers because Karpas has no bracha, and so there's no need for a kezayis; it's just a remez/zeicher, while Maror is a takana gemura to do like it was done when there was a Korban Pesach. On the other hand, Reb Moshe in the last piece in his Dibros on Pesachim does assume like everyone else in the Rosh, and klers whether the matza and maror that accompany the korban are more like the four minim by Sukkos or more like the Lechem by Korban Todah.
I just came across the Sfas Emes in Pesachim 99b, the first piece in AP, that also says this.
פרק ע ר ב י פ ס ח י ם
בגמ׳ מאי אריא ערב פסח וכו׳ קשה למ״ש התוס׳ צ"ט דמתני׳ כולל גם ע״פ שני א״כ לא הוי בכלל עיו״ט וגם באמת יש לדון ולהתיר בע״פ שני דאין לחוש אם לא יאכל בלילה לתיאבון כיון דמצות פסח הוא לאכול על השובע. רק בפסח ראשון האיסור הוא משום חיוב מצה דמצוה לאכול לתיאבון [משום הידור מצוה כמו"ש הרשב״ם לעיל במשנה בד"ה לא יאכל דלא כמ״ש התוס׳ לקמן (ק״ז:) דאכילה גסה גרע מאכילה על השובע ע״ש]. אבל בפסח שני ליכא חיוב מצה מצד עצמו אלא בשביל הפסח וא״כ נאכל על השובע כמו הפסח. וכ"מ מדברי הרשב״ם ד״ה משום חיובא דמצה כו׳ וחיובא דמצה לילה הראשון חובה דכתי' בערב ת״מ משמע דמשום דמהאי קרא ילפינן חיוב מצה בלא הפסח להכי בעי לתיאבון. אבל מקרא דעל מצות ומרורים דמשמע חיובא עם הפסח אין לחוש אם אין אוכל לתיאבון. וא״כ בפ"ש י״ל דמותר כנ"ל.
