The prayer leader on the yom tov of Rosh Hashanah — the second officiates at the blowing of the shofar [i.e., The prayer leader at mussaf officiates at the shofar blowing, and not the shacharith prayer leader. This was instituted at a time of shmad (enforced conversion), when they decreed against blowing the shofar and lay in wait for them all six hours of the time of the shacharith prayer, whereupon they moved the shofar blowing from shacharith to mussaf.]; and at the time of Hallel, the first officiates at the Hallel. [Because Hallel is not recited on Rosh Hashanah and Yom Kippur, we have "and at the time of Hallel"; that is, on the other festivals, when Hallel is recited.]
Bartenura on Mishnah Rosh Hashanah
השני מתקיע – the person who recites the Musaf prayer sounds the Shofar and not the one who recites the Morning Service Amidah prayer. And during the time of religious persecution, they ordained this, for they had decreed not to sound the Shofar, and there those lying in wait for him all six hours of the time of the Morning prayer service; for that reason, they moved the Tekiah/sounding of the Shofar from the Morning Service to the Musaf Service.
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English Explanation of Mishnah Rosh Hashanah
Introduction
On days upon which Mussaf is recited (festivals and Rosh Hodesh) there are two people who function as the “shaliah tzibbur”, prayer leader the first does Shacharit and the second does Mussaf. Our mishnah determines which of these people blows the shofar and which of them recites the Hallel.
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Bartenura on Mishnah Rosh Hashanah
ובשעת ההלל – because we do not recite Hallel on Rosh Hashanah and Yom Kippur, as it is taught, that at the time of Hallel, that is to say, on the other days of Yom Tov when we recite Hallel.
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English Explanation of Mishnah Rosh Hashanah
The one who passes before the ark on the festival of Rosh Hashanah: the second one blows the shofar. On Rosh Hashanah, the shaliah tzibbur for Mussaf is the one who blows the shofar. It is interesting that the mishnah needs to dictate this. It seems that at some early point in the development of Jewish liturgy, the Shacharit Amidah for Rosh Hashanah also included liturgy in which it would have been appropriate to blow the shofar. In our current liturgy, there is no place in the Shacharit Amidah to do so, and hence it is quite obvious that the shaliah tzibbur for Mussaf blows the shofar.
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English Explanation of Mishnah Rosh Hashanah
On days when Hallel is said, the first one recites the Hallel. On days on which Hallel is recited, the Shaliah Tzibbur for Shacharit is the one that recites the Hallel. Today we practice this by having Hallel fall in between Shacharit and Mussaf.