Transmisogyny
Transmisogyny is transphobia directly specifically at trans women. It is a combination of prejudice against trans people and sexism and/or misogyny directed at women. As such, it is necessarily an intersectional approach to the oppression of trans women. Transmisogyny can also be experienced by nonbinary people who align with or are partially women, such as demigirls or bigender people who identify as women.
History
Julia Serano made the word for her trans-feminist book, Whipping Girl: A Transsexual Woman on Sexism and the Scapegoating of Femininity (2007). By far, trans women of color are targeted by more and worse transphobic hate crimes than any other transgender group. Trans-misogyny is such a big problem because a patriarchal society-- by definition-- has contempt for womanhood, femininity, non-conformity, and gender variance. A patriarchal society sees a trans woman as a combination of all that the society tolerates the least. It is important for any activism for transgender rights to think of how their work could include trans women.
Misdirected misogyny
Misogyny is a form of sexism, specifically discrimination against women. Some misogyny ends up doing harm even to people who don't identify as women, because others see them as being like women in some way. This can happen to non-binary people, transgender men, and gender nonconforming cisgender people. That is called misdirected misogyny. Calling discrimination against men "trans-misandry" assumes the existence of "misandry," which is disputed. A patriarchal society can make misery for people of all genders, even men, but the ultimate source of that misery is the patriarchy's contempt for womanhood, not for manhood.