Chazal state that one who had sinned could not participate in war, even if he had only spoken between Yishtabach and Yotzer Ohr. The Alter Rebbe explains that Pesukei d’Zimra (passages of praise) serve as introductory praise for HaShem, and one should not interrupt between them and the rest of the davening. While Shulchan Oruch mentions permission to speak after Yishtabach (before Kaddish and Borchu) for the purpose of a mitzva, the accepted custom limits it to a pressing mitzvah, and not one that can be performed later. Thus, it is not appropriate to mouth learning or recite Tehillim at that point, and certainly not during Pesukei d’Zimra.
During Pesukei d’Zimra one may not interrupt even for a pressing mitzvah, since the brochos of Boruch She’Omar and Yishtabach must link to each other. Yet, some poskim (halachic authorities) distinguish between other mitzvos and interrupting for the sake of davening itself, and in case of need one can be lenient. If an interruption is necessary, it should be made after ‘Boruch HaShem l’Olam’ to conclude Boruch She’Omar with a brocho, and then repeat ‘Boruch HaShem l’Olam’ to begin Yishtabach with a brocho.
During Birchos Kriyas Shema and before Shmoneh Esrei of Shacharis, when one must adjoin geulah (redemption) in the prayer Go’al Yisroel, and the Shmoneh Esrei itself, it is forbidden to interrupt even for the sake of davening. Some are lenient during Ma’ariv, when adjoining geulah and tefilah is voluntary, however, the Chabad custom is not to interrupt in Ma’ariv, either. Instead, an appointed individual should make a non-verbal signal (such as banging on a table) to announce an addition, or read aloud several words of the addition during Shmoneh Esrei.
It is very worthwhile that you wish to help people learn about the prayers and have the correct intentions, so perhaps you could precede the services each time by a one- or two-minute encapsulated message that explains something about the prayers for that day, and/or send short virtual explanations.
