How to apply a custom class to a .field__item in an entity reference display?
You can add classes in the twig template directly, or through a preprocess function. 

Preprocess

Docs provide some patterns to follow: https://api.drupal.org/api/drupal/core%21includes%21theme.inc/function/template_preprocess_field/8.2.x

function HOOK_preprocess_field(&amp;$variables) {
  $element = $variables['element'];

  switch ($element['#field_name']) {
    case 'MY_FIELD_NAME':    
      $delta = 0;
      while(!empty($element[$delta])) {
        $variables['items'][$delta]['attributes']-&gt;setAttribute('class', 'custom-class-name');
        $delta++;
      }

      break;
  }
}


This applied the class to the entire entity, alongside field__item. 

Template

If you have twig debugging enabled, you can see quickly what's a good template name. I tested with field--node--FIELD-NAME.html.twig. Using the field template from classy for example, just add the class at the end of the list.

{%
  set classes = [
    'field',
    'field--name-' ~ field_name|clean_class,
    'field--type-' ~ field_type|clean_class,
    'field--label-' ~ label_display,
    'custom-class-name',
  ]
%}


This applied the class just to the field, not the label. But you could modify the template to print the classes elsewhere.