Yes you can give the improvement/betterment examination from the SAME BOARD if you fail to fulfill the JEE MAINS eligibility criteria (since your domicile is very important at the time of your admssion ie the state you reside in) and then better of the two scores will be considered and accordingly your JEE merit marks will then give you a seat in Engineering college on merit if you are able to fulfill all the eligibility criteria later.

Check the same here in the website below -

https://engineering.careers360.com/articles/jee-main-eligibility-criteria

Best wishes. Thank you.