The Sefer Kav HaYosher says the following: Taanis Esther is a day that is very auspicious for one’s prayers to be answered in the merit of Mordechai and Esther. Whoever needs mercy for any particular needs should put aside time for themselves and do the following: First, recite Chapter 22 in Tehilim. Then, pour out your heart to Hashem and ask for all your needs and mention the merit of Mordechai and Esther (whose merits saved us from Haman). The Gates of Mercy will be opened and your prayers will be accepted beratzon.
In addition to ones own personal prayers, in their own words, click here are additional special prayers for Taanis Esther/Purim from Likutei Tefilos. (Pages 100-106)
Segula for barren woman to be remembered for children
Reb Avraham of Slonim says Shabbos Zachor is a time that is very auspicious for the remembrance of barren women to have healthy living children. The reason is because it is written in Bereishis 30:22 וַיִּזְכּר אלקים אֶת-רָחֵל “and the Lord remembered Rachel” (who was barren) and it is written in Bereishis 21:1 וד׳ פָּקַד אֶת-שָׂרָה”and Hashem remembered Sarah. (who was barren). There, we had two different words of remembrance for barren woman, זכר and פקד. Now, we again have the same two words used on Shabbos Zachor. We read the maftir from Devarim 25:17 זָכוֹר אֵת אֲשֶׁר-עָשָׂה לְךָ עֲמָלֵק “remember what Amalek did to you” and right after we read the Haftorah of Shmuel-1 15:1 פָּקַדְתִּי אֵת אֲשֶׁר-עָשָׂה עֲמָלֵק לְיִשְׂרָאֵל “I remember what Amalek did to Israel”.
The following are from Segulas Yisrael quoted by Segulas HaBeShT V’Talmidov.
The Baal Shem Tov says one should get up early Purim morning and say many prayers and entreaties before Hashem for anything he or his relatives need, because the day is a very special time and all the “worlds” are happy and biratzon, and whoever sticks out his hand we give”. This refers to those collecting money, though here it refers to us sticking our hand out to Hashem asking for His help.
Reb Tzvi Hirsch of Zidachov says whoever drinks on Purim it is as if he fasted from Shabbos to Shabbos and it is a segula to sweeten harsh judgements.
Note: If you do drink, please do so responsibly and if you drink even the smallest amount
I don't know the exact source, but I have heard it said that the Purim seuda is a very special opportunity for one to pray for their needs just like Esther made her request by the seuda.
It is also brought down from the Rashash and other mekubalim that the main Purim seuda should be eaten in the morning, before chatzos. While this might not be a practical time for many, you can at least wash for some bread and have a small meal in addition to the afternoon seuda. Like this, you will cover all bases and have an extra auspicious time for tefila by having two seudas.
A Happy Purim to all. May Hashem answer all your tefilos and all the tefilos of Klal Yisrael in the Zchus of Mordechai and Esther. Amen