Hi Samiksha,
Although the rules do not mention about carrying handkerchief, the truth is the strictness of checking varies from center to center.
During my time (NEET 2016), the authorities at my center were extremely strict about not allowing anything extra including handkerchief or even a ten rupee note.
My advice to you will be, do carry it to the center if you want, ask the security person if the handkerchief is allowed when she is frisking you. She may allow after checking it thoroughly. And if she doesn't allow, you may safely hand it over to your parents.
Good luck to you!

Dear student

You can carry handkerchief with you. They did not mention about carrying handkerchief into the examhall. You may ask permission to the checker to carry kerchief by telling him your problem.

Once you are permitted you can carry handkerchief into the exam hall after thorough checking.

Hope this helps

All the best !!!