Gemara

3b - R. Chanina: In winter if one omitted mention of rain he must go back.

4a - R. Berachyah: Keneses Yisrael asked "V'Ned'ah Nirdefah Lada'as Es Hash-m...va'Yavo k'Geshem Lanu";

Hash-m: You ask that I be like rain, which is requested only sometimes. Rather, I will be like something that is always requested - "Ehyeh k'Tal l'Yisrael"!

Berachos 29a - R. Tanchum: If one forgot to request rain in Birkas ha'Shanim, he does not go back, for he may say it in Shome'a Tefilah.

This is only for one who forgot. L'Chatchilah we do not request rain in Shome'a Tefilah.

Question (Beraisa): If one forgot to mention rain in Techiyas ha'Mesim or to request rain in Birkas ha'Shanim, he must go back.

Answer #1: One who prays with a Minyan does not go back. One who prays alone must go back.

Objection: Presumably, the distinction is that with a Minyan he does not go back because he will hear it from the Shali'ach Tzibur. But R. Tanchum's reason was because he may say it in Shome'a Tefilah!

Answer #2: Rather, also R. Tanchum discusses one who prays alone;

If one remembers before Shome'a Tefilah he does not go back, for he may say it in Shome'a Tefilah. If he remembers after Shome'a Tefilah, he must go back.

Rishonim

Rif - Yerushalmi - R. Ze'ira: If one mentioned dew in winter he does not go back.

Question (Ran DH Hayah): The Bavli says that one who did not mention rain in winter goes back. Presumably, this is even if he mentioned dew. It is not clear why the Rif brought the Yerushalmi, which argues. The Ramban says that we may not add to it, i.e. to say that request for dew should suffice in place of requesting rain in winter.

The Rif brings the Gemara in Berachos.

Rambam (Hilchos Tefilah 2:9): If one forgot to request rain in Birkas ha'Shanim and he remembered before Shome'a Tefilah, he requests rain in Shome'a Tefilah. If he remembered after Shome'a Tefilah, he returns to Birkas ha'Shanim.

Ran (DH Bavli): Rav Hai Gaon says that if one remembered after Shome'a Tefilah that he did not request, he goes back to Shome'a Tefilah. The Ra'avad says that he returns to Birkas ha'Shanim, just like one who forgot Havdalah in Shemoneh Esre and forgot to say it over wine (before eating or doing Melachah, or erred in Havdalah) returns to the beginning (repeats Shemoneh Esre - Berachos 33a). The Yerushalmi says that if realized that he did not request after finishing Shemoneh Esre, he repeats Shemoneh Esre. If not, he returns to Shome'a Tefilah. This is like Rav Hai Gaon. This is unlike Havdalah, for here it is all one matter (Tefilah). Also, once he forgot in Birkas ha'Shanim he became obligated to say it in Shome'a Tefilah, so it is as if he erred in Shome'a Tefilah.

Rambam (ibid.): If he remembered after finishing his entire prayer he repeats Shemoneh Esre.

Kesef Mishneh: The Ramach says that if one prays b'Tzibur he need not return for he will hear it from the Shali'ach Tzibur. The Gemara supports this, for it says that the Beraisa that requires returning discusses one who prays alone. The Rambam omitted this, for he brings only like the Gemara says (but not what is inferred).

Rosh (1:1): Avi ha'Ezri says that if one realized that he omitted mentioning or requesting rain finishing Mechaye ha'Mesim or Mevarech ha'Shanim but before beginning the next Berachah he may say it there. This is like the Gemara says regarding one who omitted Ya'aleh v'Yavo or Retzei in Birkas ha'Mazon after Bonei Yerushalayim but before beginning the next Berachah.

The Rosh (Berachos 4:14) brings the Gemara in Berachos.

Rosh (ibid.): BaHaG says that praying again because of a mistake applies only to one who prayed alone, but if he prayed b'Tzibur he is Yotzei by listening to the entire repetition of Shemoneh Esre. The Shali'ach Tzibur is Motzi even people who know how to pray (Rosh Hashanah 34b). Even though R. Gamliel says that he does not exempt people in the city, this refers to one who did not pray at all. He is Motzi one who prayed and erred.

Tosfos (Berachos 29b DH To'oh): One who omitted an entire Berachah cannot be Yotzei with the Shali'ach Tzibur's repetition.

Poskim

Shulchan Aruch (OC 117:1): In winter one must say Ten Tal u'Matar in Birkas ha'Shanim. In Chutz la'Aretz we begin on the 60th day after the (fall) Tekufah. In Eretz Yisrael we begin on the seventh of Cheshvan.

Shulchan Aruch (5): If in winter one did not request rain and he remembered before Shome'a Tefilah, he does not go back; he requests in Shome'a Tefilah.

Bi'ur Halachah (114:6 DH b'Makom): If one remembered before finishing Birkas ha'Shanim, some say that he should say Ten Tal u'Matar and finish the Berachah from there. Others say that he can insert it where he is and continue from where he was. In all cases if one realized a mistake in the middle of a concept, e.g. after saying 'Mekayem Emunaso', he should first finish (li'Ysheinei Afar) before correcting his mistake.

Shulchan Aruch (ibid.): If he remembered after Shome'a Tefilah, if he did not (finish Shemoneh Esre and) step back he goes back to Birkas ha'Shanim. If he stepped back he goes back to the beginning of Shemoneh Esre. If he finished Shemoneh Esre and does not normally say supplications afterwards, it is as if he stepped back. If he remembered after Shome'a Tefilah before beginning Retzei, he may say Ten Tal u'Matar there and continue with Retzei.

Mishnah Berurah (6): If one forgot to request on the first night he goes back. This is unlike one who forgot Ya'aleh v'Yavo on the night of Rosh Chodesh. There, he does not go back because we do not Mekadesh the month at night. There is a minority opinion that says that one who forgot to request on the first night does not go back, just like regarding mentioning rain.

Kaf ha'Chayim (6): Even though we do not hold like Avi ha'Ezri (who learns from the Yerushalmi that in Chutz la'Aretz we wait 60 24-hour days), if one forgot to request and it is still within 60 days me'Es l'Es he does not go back. Others say that we do not follow Avi ha'Ezri at all. It seems that one does not go back at Ma'ariv, for it is always within me'Es l'Es (the earliest Tekufas Tishrei ever falls is at three hours into the night), and Ma'ariv is Reshus.

Shulchan Aruch (114:6): If one omitted Meshiv ha'Ru'ach u'Morid ha'Geshem, even if he finished the Berachah, if he did not start the next Berachah he does not go back. He mentions rain there, without a closing Berachah.

Avi ha'Ezri (brought in Mordechai 614): When one corrects the omission before beginning the next Berachah he does not close it with a Berachah. The same applies to Ten Tal u'Matar after Birkas ha'Shanim, and Ya'aleh v'Yavo after Retzei. We learn from one who omitted Ya'aleh v'Yavo in Birkas ha'Mazon on Rosh Chodesh. The Gemara was undecided whether or not he closes with a Berachah, therefore we do not.

Rebuttal (Gra DH Kodem): The source to correct the omission between Berachos is from the Rosh citing Avi ha'Ezri. The Rosh rejected this (Berachos 4:17)! Also, the proof from Birkas ha'Mazon suggests the opposite. If one can correct an omission between Berachos, why was it necessary to enact a new Berachah (for omitting mentioning Shabbos or Yom Tov)?

Mishnah Berurah (32) and Bi'ur Halachah (DH b'Lo): If omitted something for which he need not go back, such as Al ha'Nisim, once he finished the Berachah he does not say it, even before beginning the next Berachah. Many Rishonim say that it is as if he started the next Berachah, so he may not correct even omissions for which one must go back. However, most Poskim defend the Shulchan Aruch, which allows to do so. If one remembered after saying Baruch Atah Hash-m he should conclude "Lamdeni Chukecha" and correct his omission before finishing the Berachah, to fulfill all opinions. Likewise, if one remembered that he forgot Ten Tal u'Matar before starting Teka b'Shofar, he should not say it then, rather, in Shome'a Tefilah.

Kaf ha'Chayim (42): If he said Baruch Atah Hash-m he should conclude 'Mechaye ha'Mesim' and say Meshiv ha'Ru'ach u'Morid ha'Geshem within Toch Kedai Dibur.

Question (Beis Yosef DH v'Teimah): It is more reasonable to learn from one who corrects an omission of something he was obligated to mention, e.g. Shabbos or Yom Tov in Birkas ha'Mazon. Then, one mentions them and closes with a Berachah!

Answer (Beis Yosef ibid.): Perhaps there is different, for normally one closes with a Berachah (Bonei Yerushalayim) immediately after mentioning Shabbos or Yom Tov, whereas we do not normally close with a Berachah right after Meshiv ha'Ru'ach u'Morid ha'Geshem.

Rebuttal (Taz 12): Surely there was no Hava Amina to close Meshiv ha'Ru'ach u'Morid ha'Geshem with a Berachah, There are 18 Berachos, not 19 (nowadays with Birkas ha'Minim there are 19, not 20)! Also, what text would be used? Rather, Avi ha'Ezri merely proves from Rosh Chodesh that it is possible to make an insertion without a closing Berachah.

Taz (12): The Levush says that the insertion is like a continuation of the Berachah. This is wrong. Surely it is independent!

Mishnah Berurah (31): L'Chatchilah one should say the insertion Toch Kedai Dibur of the Berachah.