Rav Haim Palachi (Turkey, 1788-1869) writes that when one prays to God for something that he wants not for his own interests, but rather for the sake of God, such as for success in Torah learning, better concentration in prayer and the ability to perform Hesed, his prayer is never rejected, even if he is unworthy. Therefore, especially during the period of Aseret Yemeh Teshuba, it is worthwhile to pray for not only our personal needs, but also for our success in Torah and Misvot, and our prayers will be answered.
There are certain restrictions which many people do not observe during the year, but should be observed during the period of Aseret Yemeh Teshuba. These include "Halab Akum," drinking milk made by gentiles. Even those who do not generally make a point of drinking only Halab Yisrael throughout the year should observe this practice during Aseret Yemeh Teshuba. Likewise, those who eat "Pat Akum," bread and Mezonot products baked by gentile manufacturers, during the year, should ensure during Aseret Yemeh Teshuba to eat only Pat Yisrael (products made by Jews). This applies to all baked products – bread, bagels, pretzels, cookies, cakes and so on.
Incidentally, it should be noted the prohibition of "Bishul Akum" – food cooked by a gentile – applies throughout the year. This means that we may not eat food cooked by a non-Jewish housekeeper, even though all the ingredients are kosher and the Jew is watching her the entire time. The food may not be eaten unless a Jew was actively involved in the cooking process.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that even women who do not normally bake their own Halla on Ereb Shabbat should do so on Ereb Shabbat during the Aseret Yemeh Teshuba, so they can fulfill the Misva of Halla. They should ensure to prepare at least 3.5 lbs. of dough so they can separate Halla with a Beracha. It is preferable for a woman to separate Halla from her own dough in her home than to go to the bakery and ask the baker if she could separate Halla for him, because separating Halla from one’s own dough brings Beracha to the home. This point was made by Hacham Bension Abba Shaul (Israel, 1923-1998).
The Ben Ish Hai writes that it is worthwhile for the husband, too, to perform the Misva of separating Halla in order to accrue greater merit during the Aseret Yemeh Teshuba.
Different opinions exist among the Halachic authorities regarding the amount of dough one should separate for the Misva of Halla nowadays, when in any event the Halla is burned and not given to a Kohen. The Arizal (Rav Yishak Luria of Safed, 1534-1572) maintained that even nowadays, for Kabbalistic reasons, one should separate 1/48th of the batter for Halla. The Hid"a (Rav Haim Yosef David Azulai, 1724-1807) writes that he never saw anybody separate such a large amount of dough for Halla, but he nevertheless advises following the Arizal’s view once a year, and this should preferably be done during the Aseret Yemeh Teshuba. If possible, then, one should prepare a batter during the Aseret Yemeh Teshuba and measure 1/48th for the Misva of Halla.
Some, however, follow the practice of always separating a Ke’zayit of dough for the Misva of Halla, regardless of the size of the batter.
If a woman prepares two separate batters, and she wants to invite her friend to separate Halla from the one of the batters so she can earn the merit of this Misva, she should ensure to legally transfer ownership over that batter to her friend. This way, the friend will be able to recite the Beracha over the Misva according to all views without risk of reciting a Beracha Le’batala (Beracha in vain). This applies even if she invites her daughter to separate Halla from the second batter.
The Shela Ha’kadosh (Rabbi Yeshaya Horowitz, 1565-1630) wrote that there were certain especially pious individuals who would ensure during the Aseret Yemeh Teshuba to eat all their food in a state of Tahara (Halachic purity). Even though we are all in a presumed state of Tum’a (Halachic impurity), they were able to do this by eating only food that had not come in contact with liquid, and was thus not susceptible to Tum’a. And thus they ate baked products from flour that was kneaded with fruit juice, instead of water, as contact with fruit juice does not render food susceptible to Tum’a. For drinking, they drank only water taken from underground wells, which did not have a chance to become Tameh. This practice was observed by the Ben Ish Hai. Quite obviously, this practice is not practically feasible in our circumstances, but it is worth learning of this custom in order to get a sense of how far the great Sadikim would go to raise their standards of religious observance during this critical period of the year.
The Ben Ish Hai (Parashat Vayera) records a custom that even those who do not normally wear Tefillin during Minha do so during Minha in the Aseret Yemeh Teshuba, except on Friday.
Rav Haim Palachi writes that it is proper during the Aseret Yemeh Teshuba for parents to pray for their children and grandchildren’s spiritual success, that they should become Torah scholars and God-fearing people. The Talmud tells that Rabbi Yosef observed forty fasts, and he was then shown the verse, "Lo Yamushu Mi’picha" ("They [words of Torah] will not leave from your mouth"). He observed another forty fasts and was then shown the next phrase – "U’mi’pi Zar’acha" ("or from the mouth of your offspring"). Rabbi Yosef then observed another forty fasts, and was shown the phrase "U’mi’pi Zera Zar’acha" ("or from the mouth of your offspring’s offspring"). At that point he did not need to observe any more fasts, because once he had guaranteed that Torah would remain with him, his children and grandchildren, he is assured that it will always remain among his descendants. This incident shows us the extent to which the great Sages went to pray that they would produce Torah scholars.
The Ben Ish Hai writes that it is proper to recite the Tashlich text each day during the Aseret Yemeh Teshuba, though this is not our practice.
The Ben Ish Hai also writes that on the 8th or 9th of Tishri, one should write on a piece of paper that he wishes for all his sins to be erased from his record. He should then go to a river, list all his sins, and throw the paper into the water. The Ben Ish Hai outlines the precise details of this ceremony in his work Leshon Hachamim (2:8).
Furthermore, the Ben Ish Hai writes that it is worthwhile to recite the 24th chapter of Tehillim each day during the Aseret Yemeh Teshuba followed by the "Ribono Shel Olam" prayer for livelihood.