ANSWER: The Bible verses that correspond to your question about women covering their heads during prayer are found in the first book to the church in Corinth written by the Apostle Paul. The apostle, commenting on this issue, stated the following.
4. Every man who has a covering on his head when he is praying or prophesying puts his Head to shame. 5. But every woman who has her head uncovered when she is praying or prophesying puts her head to shame, for it is the same as being shaved.
13. You judge for yourselves. Is it becoming for a woman to pray to God uncovered? (1Corinthians 11:4 - 5, 13)
There is one VERY important fact overlooked (but implied) concerning Paul's statements regarding a woman's head being uncovered in the above verses (which also apply to 1Timothy 2:11 - 12). This misunderstanding is likely the reason why some religious groups make young, even pre-teen, girls wear a "covering" or bonnet either during church services or all the time. The Greek word translated as "woman" in 1Corinthians 11 verses 5 through 13, and verse 15, is gune (Strong's Concordance #G1135). This word refers to a female who is a WIFE. It does NOT refer to ALL women, everywhere, regardless of age or marital status! Strictly speaking, these verses concern married couples and not unmarried females!
Many people think that Paul is teaching that women MUST cover their head at all times (or at least at church) with a veil, hat or other type of covering. Some churches and fellowships also take these verses literally and enforce policies to insure their females wear a covering. These verses in 1Corinthians 11, however, are speaking more of SPIRITUAL things rather than the physical. These verses mean FAR MORE than merely putting a veil or hat on one's head.
The covering is not a hat or a veil that one wears on the head, but an ATTITUDE that a person has toward the one who is in authority over them. As Paul mentions in verse 3, God the Father is in authority over Jesus Christ, who is the Son and the Word or Spokesman (Greek: Logos), and also was the God of the Old Testament. If Christ is subject to the Father, then man needs to be subject to Christ, just as Jesus told His disciples (John 13:13).
Paul discusses in verse 5 of 1Corinthians 11 the situation where a woman (wife) prays or give prophecies with her head uncovered. But what does this mean? Paul tells us in the next verse that it would be a disgrace for a wife to cut her hair too short or shave it off. This type of action also expresses an attitude of rebellion.
10. For this reason, it is necessary for the woman to have a sign of being under authority on her head because of the angels (1Corinthians 11:10)
What did Paul mean in verse 10? Simply stated, a married woman should have long hair so that heavenly angels can differentiate between husbands and wives, just in case an angel has to quickly intervene to protect "the weaker vessel" (Hebrews 1:14). This verse has also been open to many interpretations through the centuries.
In verses 11 to 13 in 1Corinthians we are told that man is not separate from woman and vice versa. Paul also makes a rhetorical statement in verse 13 when he asks if it is right for women to pray "uncovered" or with very short or no hair. All this points to the fact that although men and women are equal in God's eye, and joint heirs of eternal life (1Peter 3:7), a wife is nevertheless to be subject to and under the authority (veiled, covered) of the man (husband). It will continue to be that way as long as there are human beings on this earth who are in the flesh.