When you fast, how do you pray ? Is there a certain prayer "culture" while fasting. Not talking about Yom Kippur or any public fast. Just fasting in general to ask Hashem for help, healing, work, support, protection, wisdom and so on.
According to the Artscroll Siddur, at Mincha on the day before this fast, a person should announce his plan to fast to Hashem with the paragraph Ribon Kol HaOlamim, in between the paragraphs of Elohy, and Yiyo Liratzon Imrei of the Mincha silent Shemoneh Esrei.
At Mincha on this fast day, a person should say Aneinu Hashem Aneinu in Shema Koleinu of the Mincha Shemoneh Esrei, and say the paragraph Ribon Kol Olamim in that same Shemoneh Esrei.
A fast day is a great time for inspecting one's deeds. I suggest meditating a little to look at all of your sins, and taking this opportunity to do Teshuva.
This OU section describes the prescribed prayers for the day of a personal fast. There seems to be no prayer "culture" while fasting but one can and should add one's personal prayers.
If a person is fasting even a private fast, whether it is a voluntary fast or a fast for a bad dream, he should say “Aneinu” (“answer us”) at mincha in the bracha of “Shema Koleinu,” the same as one would do on a public fast. Even though he is an individual, he recites the prayer using the plural (“us” rather than “me”) so as not to change the wording that was established by the Sages. Before “Yihiyu l’ratzon,” he should say “Ribon ha’olamim.” (See OC 565:4 for the text of this prayer.)
Here is the text of the prayer (block letters without vowels at the bottom of the page and the next page of the linked document)
רבון כל העולמים גלוי וידוע לפניך בזמן שבית המקדש היה קים אדם חוטא, ומביא קרבן ואין מקריב ממנו אלא חלבו ודמו ואתה כרחמיך הרבים מכפר ועתה בעונותינו חרב בית מקדשנו ואין לנו מקדש ולא כהן שיכפר בעדנו לכן יהי רצון מלפניך שיהא מיעוט חלבי ודמי שנתמעט היום כחלב מונח על גבי המזבח לפניך ותרצני
Lord of all worlds, it is revealed and known before You that when the Temple stood, a man could bring a sacrifice and only offered up its blood and fats and You in Your great mercies wiped away the sin. And now because of our sins our Temple is destroyed and we do not have a sanctuary not a Cohen to get atonement for us. Therefore may it be Your will that the reduction of my fat and my blood today should be as fat put on the altar before You and accept me.
You also find on that page the prayer to be said at the mincha before the fast day
(starts רבון כל העולמים הרי אני לפניך בתענית נדבה ).