WordPress ajax success response

You can’t use 2 echos for an ajax response.

Try this:

js:

  $atj.post(MyAjax.ajaxurl, requestData).done(function(result){        
    console.log(result);
    if(result == 'success'){
      $atj('.training-form [type=text]').val('');
      $atj('.training-form-message').append('<p class="training-form-complete-message">Thank you for the  email</p>');
    }
  });

php

function myajax_submit() {
// snip...

 $mail_check = wp_mail( $to, 'Email Test', $message );
 remove_filter( 'wp_mail_content_type', 'set_html_content_type' ); 

 if( $mail_check) 
   echo 'success';
 else
   echo 'failure';
 exit;
}