Mentioning mental health in a supplementary application essay I'm applying for a PhD in neuroscience and I've recently finished writing my SOP and supplementary essay. My supplementary essay asked me to respond to the prompt "Please tell us about an incident in your research where you experienced failure. Include what you did about it and what you learned from the experience." I wrote about how due to my social anxiety I failed to reach out for help in preparing for graduate school, leading to a delay in attending graduate school. I wrote about how I corrected my behavior to no longer respond to anxiety with avoidance. Most of the essay focuses on what I did to correct my behavior and my talk of mental health is kept fairly minimal. I asked the admissions whether it was okay to write about a personal difficulty I overcame throughout my academic career instead of a technical failure in the research process, and they said this was fine. However, when I showed my essay to my academic advisor he said I should avoid mentioning mental health and therapy as it might scare off the admissions committee. The specific lines he had a problem with were as follows: 


  "However, towards graduation I became preoccupied with moving away
  from my parents instead of preparing for graduate school. This was
  despite the fact that they offered to pay for my graduate education on
  the condition that I stay. I later learned, through working with my
  therapist, that I was attempting to escape an abusive environment."


I feel as though these lines are very important to contextualizing the situation. Would a line like this really scare off admissions?
I'm pretty sure you misinterpreted the question. I would think that they wouldn't (dare) ask about your personal failings or mental state. The question as I interpret it is about a setback in the research itself. Suppose you don't get enough subjects to properly carry out your experiment as proposed. What do you do. Or you learn something part way through an experiment that invalidates it and you have to go back and revise. 

If you never have setbacks then I'll guess that you aren't trying hard enough to get to the important questions. For a mathematician to be able to prove every proposed theorem says those theorems aren't very consequential. (I've been in that place, actually). 

Focus on the research, not your own psychology. Your application should clearly indicate that you are absolutely the best person for this position and will obviously be a success. Moreover you are flexible in overcoming the unknowns that pop up during the execution of it.