artachshasta: It’s incorrect that we skip from Misod to Zachreinu on second day mussaf. We don’t say Misod AT ALL on second day mussaf since no piyutim are said in first brachos! Check any traditional machzor. As such, this proves the point that Misod is NOT said if we skip to Zachreinu, and if Misod is said, it should be followed by the piuytim it is asking for permission to recite.
rebdoniel and others: Many mention that the Gra did not recite piyutim. PLEASE PROVIDE A SOURCE THAT IS BETTER THAN MAASSEH RAV. Masseh Rav is VERY CLEAR that the GRA said ALL of the piyutim in chazaras hashatz on RH and YK. Please look it up inside (in the section of RH and YK) and then get back to us here, instead of making things up (though it is a common error amongst yeshiva educated people who don’t bother to open up maaseh rav or offer a better source). The Gra did not say the piyutim in bircos krias shma. Also, on other Chagim, the Gra held that the piyutim were hefsef, but still said MANY of them (including ve’ata bonim on shavuos, bracho dodi on pesach….) AFTER HALLEL. As someone who is worried of hefsek but tries to follow the Gra I hope you also say these beautiful piyutim after hallel (the Rav by the way did the same thing), as well as the yotzros of Arba parshios, like the Gra and the Rav, after kaddish.