Should I apply with a 506 and planning to retake?

So I scored a 506 from my April 10th exam. However, I have been testing 513-516 on all my practice tests, and I scored a 512 on a practice test 2 days after I got my scores back. Should I apply now and mark that I plan to take the exam again in late June so that my application can be processed? How does this work? Is it possible some schools will discard my application before seeing my new score?

Thank you for any advice!

to be honest, i would still apply IF the rest of your application looks good, i.e. great GPA, experiences, essays, etc. Apply to your state schools, DO schools, and use msar to find schools where you come in the 10th percentile. I would actually advise you to put your dream schools also on your list if you have a solid application as you never know. Worst comes to worst, retake your exam if you don’t get in this cycle.