Insert custom content before widget title/after widget opening tag

You can use the dynamic_sidebar_params filter:

add_filter( 'dynamic_sidebar_params', 
    function( $params )
    {
       // target a sidebar id:
       if( isset( $params[0]['id'] ) && 'sidebar-1' === $params[0]['id']
       {    
           // target a widget name:
           if( isset( $params[0]['widget_name'] ) && 'Text' === $params[0]['widget_name'] )
           {
               // target a widget id:
               if( isset( $params[0]['widget_id'] ) && 'text-8' === $params[0]['widget_id'] )
               {
                   // Append text to existing 'before widget' markup code:
                   if( isset( $params[0]['before_widget'] ) )
                   { 
                       $params[0]['before_widget'] .= '<!-- Custom content -->';
                   }
               }
            }
         }
         return $params;
    } 
);

where the widget names are for example: Text, Categories, Calendar, … .

You can then modify it to your needs, for example the 'sidebar-1', 'Text' and 'text-8' parts.