In John 4:24, Jesus is making a contrast between the worship of the Jews and the worship that God was going to ask of His followers under His new covenant. But, the Jews must also sincerely worship (Daniel 6: 4-7; Isaiah 1: 10-18) and according to the truth (Daniel 5: 22,23). Therefore, to say that to worship sincerely and according to biblical authority is the main emphasis of the text, is to ignore the contrast that Jesus has in mind.
Worship in spirit: Means worship of a spiritual nature in contrast to worship with the material emphasis of the Old Testament.
The adoration of the old covenant was designed to call attention to the physical sense of man and therefore had many material accessories: the tabernacle, special clothes for priests, lamps, incense, instrumental music, animal sacrifices, etc. (Heb 9: 1-10) New Testament worship has a spiritual emphasis (1 Cor 3:16, 1 Peter 2: 5,9, Romans 12: 1, Heb 13:15, etc.), that is, it is more about the spirit of man.
Worship in truth: It is worship based on the actuality of Christ in contrast to that based on shadows. The props of Old Testament worship were shadows, that is, anticipated the true worship of the New (Hebrews 10: 1). For example, the tabernacle was a shadow of the church and heaven (Hebrews 9: 8,9), circumcision was a shadow of baptism (Colossians 2: 11,12), animal sacrifices one of the sacrifice of Christ, etc. But when Christ arrives, we no longer worship on the basis of shadows, but on the basis of reality, the sacrifice of Christ on the cross, our salvation from sin and the living hope of heaven. We no longer need incense because we rely on Christ as Mediator in prayer, we no longer worship with instrumental music because we want to emphasize the heart by giving God the fruit of our lips (Hebrews 13:15), we no longer need animal sacrifices to point out Something to come, because Jesus came and made the sacrifice that is effective "once for all."
This text contradicts those who want to maintain the physical elements of Old Testament worship, for example, incense, instrumental music, etc. and also to those who do not want to release the shadows like the tithe, the Sabbath, etc. Let us leave the adoration of the Old Testament and adore God in spirit and in truth.