I completed my Masters in mathematics in my home country (a developing country in Asia) in June 2020. I applied to 6 professors to do a PhD in the last semester of my Masters. I was rejected for 5 and funding couldn't be arranged for the 1 in which I was accepted. So I thought I would try next year in both my home country and abroad (Europe). I didn't apply earlier to any of my home country's universities.
So, I started studying for the qualifying exam in my home country while side by side studying some more to apply for PhD programmes abroad. I took the exam for PhD admission in my home country in December and received the result in early January. The result was lower than I had hoped for, although I don't yet know if I've been accepted.
My father is using this to make cruel comments to me, hurting my mental health. This contributed to me being depressed and missing several application deadlines for European universities. I completed only one application on time. Since then I have received medical help for my mental health problems.
How should I explain why I didn't apply to other programmes to the admissions committee of the university I did apply to ("university X")? Will it hurt my application if they know my father is against my decision or that I have had mental health problems? Should I lie (e.g. say I had COVID)?
If I get an interview call from university X can I request to be interviewed later than other students so I have time to prepare?
I am really passionate about research and also returning to academia after quitting would be really hard for me. I don't want to do a PhD in my home country because my experience is that the quality of teaching is low.