It’s a segulah, NOT a minhag. The segulah is to have a GIRL and has nothing to do with an easy childbirth. Bas techilah siman yafe lebanim, guarantees you three children minimum over the course of life. So, it’s worth the try!
There is NO Yehi Ratzon, you can add your own heartfelt tefillah that all should go well. It’s like the Yehi Ratzon to have a raise in salary on Rosh Hashanah night!!