"Genome Harassment" or "Genohara" is a reference to one of Fuji Kiseki's training events.
Originally (pre-Umamusume; I'm getting this from the Pixiv wiki), the term was used to refer to discrimination/bullying based on genetics - making fun of people with genetic diseases for example.
In Fuji Kiseki's case, it's a shortening of "Genome Sexual Harassment". In her training event "The Happiest Thing in Life (人生最大の幸せとは)", after what appears to be some exercise in the park, she borrows the trainer's towel, mentions that it smells comforting, that she's heard that thinking that someone smells good means that you're compatible gene-wise, and then asks if the trainer thinks she smells nice too.
In other words, this is a legitimate investigation to find out if Trainer-dono is in any danger..."A case has got a unique smell to it", is what my dad said...That's right..."Genome Harassment" is justice...And "you've gotta follow your nose when you're investigating a case"...is what my dad said
