I would like to make a few enquiries about my application.

  1. I have a couple of enquiries regarding my application.
  2. I would like to make a few enquiries about my application.

Are the above sentences correct? Please advise.

They are correct. If it is something formal, I would use number 2 given the choice of those examples. If given a free choice I would use this:
I have some queries regarding my application.