The Shulhan Aruch (O.H. 112:2) rules that one may not insert liturgical poems (Piyutim) and the like into the prayer. The Rama, however, says that there are opinions which do permit their recital because there is a communal need for them. In this context, the Shulhan Aruch is referring to the central part of the prayer, that is, the blessings of the Shema and the Amida. Although the Ashkenazic rite includes several examples of Piyutim that are inserted into the prayer, it is less common among Sephardic communities and is usually only done during Rosh Hashana and Yom Kippur. Rabbi Haim Ben Attar (Hefetz Hashem, Berachot 11) says that inserting Piyutim in the prayer is a longstanding custom that predates the Shulhan Aruch and that it should be firmly upheld.
There is a more common custom, especially among Moroccan Jews, to insert Piyutim in the Pesuke Dezimra.These Piyutim are recited not only on Rosh Hashana and Yom Kippur, but throughout the year, such as Shabbat Shira, and during life cycle events, such as a Brit Mila.The HIDA (Tov Ayin, § 18:35) questions how opponents to this custom could find any wrongdoing with it. The whole point of Pesuke Dezimra is to sing praises to Hashem and by inserting other Piyutim, it is only enhancing this praise. The famous Piyut “Mi Kamocha” by Rabbi Yehuda Halevi, which is recited on Shabbat Zachor, ends off with words that are clearly connected to Nishmat, which proves that it was meant to be sung at that point. Furthermore, the prolific Paytan and scholar Rabbi David Hassin ended many of his Piyutim with the words “Nishmat Kol Hai” to indicate that their proper place was in Pesuke Dezimra. Rabbi Yehuda Toledano (Vezot L’Yhuda) confirms that this is the Moroccan custom and that anyone who opposes it violates the will of the Rishonim. Rabbi Shalom Messas (Shemesh Umagen, vol. I, § 41) and Rabbi Avraham Adadi (Vayikra Avraham, pg. 122) concur and add that not only are these Piyutim not an interruption, as some contend, but rather they enhance the awe that one should have when praying.
Summary: Reciting Piyutim in Pesuke Dezimra is a legitimate custom and is not considered an interruption.