A timed application is timed for a reason. Not necessarily a good reason. But if they wanted to give you unlimited time so that you could formulate your thoughts with maximum clarity, they would have done so. They didn't.
They were instead looking for what you could do in the time allotted, just like all other applicants. At best, they wanted to see what you could do within that amount of time, or possibly how you handle deadlines on these sorts of tasks-- perhaps writing under pressure is part of the job.
Emailing an explanation/excuse for what you imagine to have been a poor performance might send good signals:
JeffM understands his limitations, and is aware of the quality of his output and seeks to correct/improve where necessary. That's great!
It might send bad signals:
JeffM performs poorly under time pressure and then is panicky afterwards. We can't have that when deadlines are approaching!
It might send no information at all:
This email is from a recent applicant, and the subject line says "explanation of application issues". I'm not going to read it.
I doubt it would harm you to reach out, but I don't see how it could do much good for you either. Leave it be.