Head Covering for Women - needed for Beracha?

 https://ph.yhb.org.il/en/03-10-06/

Reciting berakhot is not considered to be standing before the King, so a woman in private, when the laws of tzni’ut do not require her to cover her hair, there is no need for her to cover her head while reciting a berakha. However, Yabi’a Omer 6:15 and Halikhot Bat Yisrael 5:3 state that married women must cover their heads when mentioning God’s name, even when in complete privacy. Still, in practice, many married women normally recite the bedtime Shema and Ha-mapil without a head covering. It seems that the reason for this is what I wrote above, that the obligation to wear a yarmulke when reciting berakhot is based on the pious custom to cover one’s head throughout the entire day. Since the yarmulke inspires fear of God all day, it follows that one must also wear it while invoking God’s name. However, women, who do not customarily wear a yarmulke to inspire fear of God, need not wear one when mentioning God’s name either. There is proof for this in m. Ĥalla 23: “A woman may sit and separate the ĥalla while nude.” Nude here obviously means bareheaded, for it would not make sense that she is required to cover her head but allowed to recite the berakha while nude. R. Shlomo Zalman Auerbach rules this way too, saying that women’s custom to cover their heads in the mikveh while making the berakha on immersion does not obligate them in this practice for all other berakhot (Halikhot Bat Yisrael 5 n. 6). Therefore, a woman who wakes up in the middle of the night to relieve herself is not obligated to cover her head while reciting Asher Yatzar. Likewise, a woman who walks around her house bareheaded (when there are no strangers around) and wants to drink something is allowed to recite She-hakol and Borei Nefashot without a head-covering.