How to override default WordPress core language file

Correcting the answer given by Max. To use this approach, the filter you need is gettext_with_context because your string is translated via _x() not __().

Note that the filtered argument passed is the translation, so you should check the source string ($text and $context) to ensure you’re filtering the correct item.

add_filter( 'gettext_with_context', 'added_translation', 20, 4 );

function added_translation( $translation, $text, $context, $domain )
{
  if( 'Aside' === $text && 'Post format' === $context ){
    $translation = 'Aan de kant';
  }
  return $translation;
}

Up to you exactly how you detect the string and whether you also check to see if Dutch is the current locale. However this filter does work and should work regardless if the theme you’re using.

If you’re making a lot of string improvements I recommend using a translation file loaded in addition to the installed translations. Example guide here using Loco Translate to add your own custom file. (Disclosure: my plugin)