I'm sure that many students will understand the struggle of reaching the word count, it can be quite tricky sometimes! I hope you are not feeling too panicked.
On my course at university (York St John), we have the policy where we have to be within 10% of the word count. This means that if I am below a certain word count or above, then my mark will be penalised (usually capped). They have a word count for a purpose so you should always try to reach it. If I were in your position and at the time needed 6 extra words to be within the word count then I would have tried to fit them in somewhere. Obviously every university will approach things slightly differently so you could ask a tutor what the rules are around this.
To try and increase my word count I like to do all of my reading before beginning writing the assignment, and finding research from all the different areas that I need to look into. I don't start writing until I feel that I have researched enough to comfortably know what direction my essay would be written in. My uni said that as a general guide, you should be using at least one reference per 100 words in the assignment. It's much better in my opinion to have written too much and work on cutting it down than at the end of writing trying to find more things to talk about.
I don't think you would fail for a shorter word count, but it has to be mentioned that the reason they ask you to write 2000 words is because your essay should be in depth enough to sufficiently cover 2000 words worth of research.
I wish you the best of luck with your future assignments and hope that helped!
Rachel- YSJ student rep