Which sects of Christianity recite verses of God and prostrate to God during the night?
(This post was made prior to to edits of the original question.)
As indicated in your Wikipedia link, many Christian denominations practice prostration at various religious services. However, this could be more of a local tradition, within a particular denomination and region where a true set of norms and rules do not exist because it is more or less a private devotional act.
In any case, Christians are not generally in the habit of praying uniquely in the name of God without associating the every essence of God with the name of the MostHoly Trinity. We are Christians after all and the Sacred Trinity is God.
There is a tradition within the Catholic Church in Poland where the polish faithful pray in a prostrate manner. Since it is a personal devotional manner of praying the Church does not regulate what prayers or biblical verse must be recited. Nor does the Church regulate times in which the faithful must pray in this manner.
Pope St. John Paul II often prayed prostrate on the ground, according to Polish tradition.
Throughout his life, from the first moment of each day to the last, John Paul kept a regimen of deep, fervent devotions that left those around him, even other religious leaders, in awe.
As he said in the early days of his papacy: "Prayer, which in so many ways expresses our relationship to the living God, is the pope's first duty and his first message, the first condition of his service to the church and the world."
During his Vatican years, health permitting, John Paul would spend as many as seven hours a day in prayer, adoration and contemplation. In "Great Souls: Six Who Saved the Century," journalist David Aikman described the routine during the pontiff's prime:
"John Paul II is at prayer in his private chapel by 6:15 a.m., at times prostrate on the floor, at times actually groaning in the travail of intercession. During moments of down time' amid the panoply of solemn public appearances, he again and again closes his eyes, tightens his facial muscles in concentration, sometimes shades his eyes with his hand, and withdraws to that inner, sealed room of his soul where he communicates with God." - John Paul's fervent prayer inspired awe in others
I am sure many Christians around the world pray in such a manner, but I know of none that regulate it with specific norms outside of very specific religious ceremonies. Thus generally, it would be a private devotional act that some Christians practice privately as they do in Poland. The prayers of biblical verses recited would naturally be left up to the individual.
I see no need why such private devotional acts should be regulated. The Holy Spirit can inspire the faithful to pray the way He desires.
Remember also that Jesus himself prayed in this position, just before his Passion.
"And he (Jesus) went a little farther, and fell on his face, and prayed, saying, O my Father, if it be possible, let this cup pass from me: nevertheless not as I will, but as thou wilt." - (Matthew 26:39)
Whether or not some nontrinitarians prostrate themselves at night and invoke the name of God, but not the Trinity seems to be regulated by private devotion also as I can find no source stating rules for such an external observance.
Many nontrinitarians are generally not recognized by Trinitarian Christians as many dream their baptisms as invalid.
Nontrinitarianism is a form of Christianity that rejects the mainstream Christian doctrine of the Trinity—the teaching that God is three distinct hypostases or persons who are coeternal, coequal, and indivisibly united in one being, or essence (from the Greek ousia). Certain religious groups that emerged during the Protestant Reformation have historically been known as antitrinitarian.
According to churches that consider the decisions of ecumenical councils final, trinitarianism was definitively declared to be Christian doctrine at the 4th-century ecumenical councils,13 that of the First Council of Nicaea (325), which declared the full divinity of the Son,5 and the First Council of Constantinople (381), which declared the divinity of the Holy Spirit.6
In terms of number of adherents, nontrinitarian denominations comprise a small minority of modern Christians. The largest nontrinitarian Christian denominations are The Church of Jesus Christ of Latter-day Saints, Oneness Pentecostals, Jehovah's Witnesses, La Luz del Mundo and the Iglesia ni Cristo, though there are a number of other smaller groups, including Christadelphians, Church of the Blessed Hope, Christian Scientists, Dawn Bible Students, Living Church of God, Assemblies of Yahweh, Israelite Church of God in Jesus Christ, Members Church of God International, Unitarian Christians, Unitarian Universalist Christians, The Way International, The Church of God International, and the United Church of God.
More information may be gleaned from the following articles: