No they don't expect it to be perfect! They just want to see what your current work is like and what your teacher thinks about it (they usually want teacher comments I think). Tbh they may even think it is better than the teacher has said; marking essays is subjective and I doubt your teacher has been wholly accurate when trying to rush through a pile of classwork.
Also, remember that it is not the only part of your application! There are many other factors to take into account; GCSE, AS levels, predicted grades, personal statement, outside reading, teachers reference, admissions test, etc. Having one part of your application that is not quite perfect will not matter.