Is it permissible to recite the order of confession printed in most Rosh Hashanah prayer books between the sets of the shofar blasts?
It is a positive Torah commandment to hear the shofar blasts on the day of Rosh Hashanah, as the verse states: “It shall be a day of [shofar] blasts for you.” One may not speak between the various sets of shofar blasts, and certainly not during the blasts themselves.
The Poskim disagree regarding whether or not it is permissible to recite the order of confession.
Rabbeinu Ovadia Yosef zt”l rules that one may not recite this confessional prayer between the sets of the shofar blasts. Doing so causes one to enter into the realm of saying a blessing in vain, and questionably one may need to recite the blessing on the shofar blasts again. Even if one has customarily recited this confessional prayer in the past, one must discontinue this custom at once.
Nevertheless, it is permitted if some one silently read this confessional prayer to themself and ponder it in their heart between the sets of blasts. However, during the time the shofar is being sounded, they must not only remain silent, but listen carefully to the shofar blasts.
Some people follow the ruling of the Arizal and Shlah, and they do recite the order of confession between the sets of the shofar blasts. Where a congregation follows this custom, the shofar blower stops blowing in order to accommodate them. (הגריש"א והמנחת אלעזר ודבר יהושע)
Therefore, both practices are justified.
In contrast, it is never permitted to insert a prayer between the recitation of the beracha and the beginning of the teki’ot (shofar blasts).