Romans 10:9,10 For if thou shalt Confess with thy mouth the Lord Jesus and believe in thine heart that God has raised Him from the dead, thou shalt be saved, for with the heart man believeth unto righteousness and with the mouth confession is made unto salvation.

That can become a cathartic prayer for salvation.

The Psalms of David are filled with cathartic prayers. #s 55 and 56 are just such examples.

The Prayer of Job, in the last chapter, after God has responded to him, can be such a prayer.

Psalm 6 says <font color="purple">[b] O Lord rebuke me not in thine anger, neither chasten me in thy hot displeasure. Have mercy upon me, O Lord for I am weak, O Lord heal me, for my bones are vexed. My soul is also sore vexed; but thou O Lord, how long. Return O Lord deliver my soul, oh save me for thy mercies sake. For in death there is no rememberance of thee, in the grave who shall give thee thanks? I am weary with my groaning; all the night make I my bed to swim; I water my couch with my tears....

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It isn't the actual words that are said that make it carthartic. It's the "vexation" that preceeds it and the cleansing feeling/knowledge that God has heard once made, imo.