Shalom Rav I see many women pray without any handkerchief on their heads, but why do others put on a handkerchief? If a man must cover with a kippa (yarmulka, head covering) for prayer, why isn't the women obligated to ?

You agree with Rav Ovadia Yosef who obligates women, just like men. You can obviously do so, but there is no obligation. To understand those who are lenient, it may be an historical reason, that girls used to marry relatively young at about 12, and accordingly for most of their lives, they wore a head-covering fulltime as dictated by the Torah. On the other hand, those still unmarried were also under the age of obligation of mitzvot. Others explain that the goal of the kippa (head covering for men) is to stress awe of heaven (Shabbat 156b), which is accomplished by women even more so by totally dressing their entire bodies according to the laws of modesty, so there is no need for just a head covering. Accordingly, the custom of the unmarried women is not to need a special head covering for prayer.