Q Can you make Shecheyanu on jam made from Ethrogim on the second night (of Rosh Hashonoh)?

A. The Tur and Bais Yosef (O.H. 583) quote from Talmud Kerissos (6a) that one should eat on Rosh Hashonoh esrogim as a good siman. (In our text-version of the Talmud the word esrog is omitted). Ben Ish Chay (Parshas Reeh) mentions that it was a Yerushalmi tradition to eat esrogim on the second night and recite shechecheyonu on them. He adds, that in his city, the esrogim were indeed sweet, but hard to find before Sukos. The ones they did eat were the ones left over from last Sukos and were consumed on Tu Bishvat, but no shechecheyonu was said, since that brocho was already recited during Yom Tov on them. A similar opinion is to be found in Leket Yosher (p. 149) and in Hoelef Lecho Shlomo (O.H. 92).

A number of other reasons are mentioned by Poiskim not to recite shechecheyonu on a esrog. Mishna Berura (225:16) quoting Shaar Ephraim and Mor Uketzio, explains that since the esrog fruit remains in its tree from year to year there is no recognizable pleasure in its renewal. Sdei Chemed (Brochos 2:2 in the name of Yofe Lalev) reasons that since the esrog is also compared to a vegetable (Bikurim 2) no shehecheyonu is said.
If the esrog was also preserved as a jam there are numerous Poiskim who opine that the blessing is omitted, since it can endure for many years in that state and its beginning will remain unidentifiable. However, some Poiskim maintain that if the esrog is fresh and it was not used on Sukos, shechecheyonu can be recited as was the Yerushalmi minhag quoted above. (Tehilo Ledovid p.153 – Eitz Chaim, Hilchos Suko p.352).

Horav Shlomo Miller’s Shlit”a opinion is that you can use a new fresh esrog (not preserved) on the second night of Rosh Hashonoh, for reciting shechecheyonu by the kidush.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a