The key is whether you can realize your own problems from this matter and learn to be cautious in everything in the future.
The job is not easy. You can't ask recruiters to look at hundreds of resumes every day and take special care of you.
Of course, it is ok to send another letter, provided that you can sincerely apologize instead of just explaining it. Don't expect the second letter to add a lot of points to your job application.
Whether this company can finally admit you or not, this is a lesson you must face.