In Eph. 6:18 Paul instructs believers to “pray in the Spirit”, so the doctrines of the HS and Prayer are closely related.
Jesus taught believers to pray to God “our Father in heaven” (in Matthew 6:9a, cf. Luke 11:2-4). Addressing God as our heavenly Father connotes that we who pray are children of God, reborn of the Spirit (John 3:3-8) through union by Faith with God’s Son (Rom. 8:9-17, 1John 5:1-6).
James 2:19 says demons or ungodly souls may believe in God’s existence, so perhaps they might address God in order to curse Him (Job 2:9, cf. John 17:1).
Jesus prayed for the Kingdom of God (KOG) or God’s will to be done on earth as it is in heaven, referring to the reign of His Spirit in the hearts of believers (Luke 17:20b-21). The KOG is manifested in the world as salt and light or as love and truth (cf. TOJ #19), which is the HS (Rom. 5:5, 1John 5:6). Examples of reflecting the HS’s love are mentioned in the Parable of the Sheep and Goats (Matt. 25:31-46, TOJ #35).
In the writings of Paul, the KOG is called the body (1Cor. 12:13, Eph. 1:23 & 5:30) or church of Christ (Col. 1:13 & 18, Eph. 2:22, 3:10 & 5:23), into which believers are baptized by the HS (1Cor. 12:13).