Which English form is correct:
do you have any updates or do you have any update ?


do you have any update

It's not correct. Don't use this phrase.

It can be difficult to match the word 'any' with the right plural or singular form of a word. Usually, if you can count an item, then it should be plural when paired with 'any':

  • Do you have any pets? Yes, I have 1 cat.
  • Did you buy any books? Yes, I bought 10 books.
  • Did you see any movie stars? No, I did not see any movie stars.

But if it is something that would be difficult to count, then you use the singular form:

  • Do you have any salt? Yes, I have some salt.
  • Do you have any water? Yes, I have water.
  • Did you eat any meat? No, I am a vegetarian, I did not eat any meat.

This is a question asked to learn if there is new information about something.