1 Corinthians 11:10 New English Translation

¹⁰For this reason a woman should have a symbol of authority^[a] on her head, because of the angels.^[b]

1 Corinthians 11:10 sn Paul does not use a word specifying what type of “covering” is meant (veil, hat, etc.). The Greek word he uses here (ἐξουσία exousia; translated symbol of authority) could be (1) a figure of speech that may substitute the result (the right to participate in worship) for the appropriate appearance that makes it possible (the covered head). Or (2) it refers to the outward symbol (having the head covered) as representing the inward attitude the woman is to possess (deference to male leadership in the church).
1 Corinthians 11:10 sn Paul does not explain this reference to the angels, and its point is not entirely clear. It seems to reflect an awareness that angels are witnesses to church life (cf. Eph 3:10) and would be particularly sensitive to resistance against God’s created order.

New English Translation (NET)