Rebbe Nachman of Breslov has many specific teachings on prayer. He taught that one should speak to Hashem in one's own words for at least an hour each day, thanking and praising Him, doing teshuvah, asking to come close to Him, and asking for any material or spiritual things we need. Though the main emphasis in much of his writings is on praying about spiritual matters, Rebbe Nachman also taught that one should pray for any physical thing one needs, even something as insignificant as a missing button on one's coat. Here is one paragraph partially summarizing his approach:
"It has already been explained how important it is to seclude yourself and pray, and how powerful a method this is. It is the path by which we can come close to God. Everybody should set aside fixed periods every day and express himself before God in his own native language. It is much easier to say what you need to say when you are using your own language. You should set forth whatever is in your heart. Use every kind of appeal and argument. Use words that will endear you to God and win His favor. Plead with Him to draw you closer. Every individual knows his own personal pain and sorrow and the distance that separates him from God. It is impossible to convey the true greatness of this method. It is superior to all others. It is the way of serving God, and through following it everyone can attain the ultimate good in this world and in the World to Come. There is nothing that cannot be accomplished by prayer and entreaty. The greatest of the Tzaddikim achieved what they did only through this practice. Think about it carefully and you will see the greatness of this path. Set aside one hour every day for this, the rest of the day be happy -- and then you will be truly blessed." (Likutei Eitzot, Hitbodedut).
More can be found here: http://www.azamra.org/Advice/meditation.html and here: http://www.azamra.org/Advice/prayer.html and here: www.azamra.org/Essential/hisbodedus.htm
Rabbi Shalom Arush, a Breslover Rosh Yeshivah in Israel, has two additional teachings regarding praying for a particular thing. If one is in need of something very important, he teaches that one should pray for six hours straight for that thing, or for an hour a day for that thing for a whole month. He reports many miraculous salvations resulting from this practice, as described in his books and articles.