Video interview fail

I took part in a video interview a couple of days ago. it was a pre-recorded interview that required me to answer a few questions.

However, i felt that my interview wasn't great. i had just received bad news about a family member passing away that morning, but I had to do the video interview because the deadline was on the following day.

I was able to answer the questions, and had prepared before hand. But it feels like my energy that i wanted to portray and my eagerness to work was clouded by my family news. I am scared that I've missed an opportunity to bag my dream graduate job.

The last thing i want to do is make excuses, but I'm thinking of emailing them to explain my situation. The next stage of the application process is a face to face interview, which i believe i will be able to do much better in.
I believe it was just the timing of the whole situation that threw me off, and i don't want to miss out on the shortlisting without a fight!
I have a genuine interest in the company, and have even completed a short internship with them last year.

I have not received any feedback from my pre-recorded interview, and was thinking of emailing them before they let me know.

Do you think this is a good idea? and if so, how would you suggest i structure my email, to not sound too much of an excuse.

Yeah but be professional