Question: Last Rosh Hashana I went to a
different shul to normal. I started saying the yehi ratzon
printed in my machzor between the tekios but someone motioned for
me not to say it. Can I say it if I want to?
Answer: The Shulchan Aruch (OC 592:3) writes that one should not talk between the various tekios. The Tur (OC 592:2) writes that while one who spoke would not need to repeat the berachos and listen to the shofar again, nonetheless, they should be told not to talk. The Rema adds that davening and shofar related speech does not constitute a hefsek, unnecessary interruption.
R’ Benzion Abba Shaul (Ohr Letzion 1:39) maintains that there is no hefsek to recite the yehi ratzon, being that according to the Arizal, such words of vidui are appropriate during the blowing of the shofar. Similarly, the Mateh Ephraim (590:36) records the minhag to recite the yehi ratzon, though cautions against reciting the names of the malachim, angels.
Nonetheless, the Mishna Berura (592:12) writes that the Rema is specifically referring to the tefillos in between each set of blasts. One must not make such interruptions in the middle of a set, however. Therefore, one should not recite the yehi ratzon that is printed in the machzor. He notes (Shaar Hatziyun 592:15) that R’ Yaakov Emden (Siddur Beis Yaakov) was lenient in this regard. If one is in a place where the minhag is to recite yehi ratzon, one should not stop them.
R’ Ovadia Yosef (Yabia Omer OC 1:36:18; OC 3:32) quotes the Minchas Elazar (1:75) who challenges the minhag to recite them. The Beis Yosef (OC 590) writes that there is a machlokes as to why we must blow a minimum of thirty blasts. According to Rambam (Shofar 3:2), the reason is because there is a safek (doubt) as to which is the correct sound for teruah. Accordingly, we blow three sets to ensure that we fulfil the mitzva. Therefore, R’ Ovadia argues, it would be a hefsek to interrupt with any tefillos in the middle. Additionally, there are some unsavoury names that have made their way into the text which must not be uttered (See Minchas Elazar 1:75).
R’ Shmuel Wosner (Shevet Halevi 5:65:3) and R’ Moshe Sternbuch (Teshuvos Vehanhagos 1:344) note that many great Rabbis never said this yehi ratzon.
In conclusion, unless one has a specific minhag to say the yehi ratzon, it is best not to recite it.