Can an STD be passed from contact of hand to genitals?

I had a one night stand (my first and last one) 5 nights ago. We used protection but now I’m worried that I may have an STD because I’m extremely itchy down below and I also have thick white discharge… I had thrush over a month ago but I took a tablet for that and it disappeared… The itching and white discharge is exactly what I had when I had thrush, so I don’t know if I’ve got it again or this time I have an STD.

We used protection but I think he may have touched his penis and then touched my vagina with his hand… What I want to know is, can STDs be passed from hand to genitals?