Filter the Event Espresso upcoming events widget so it only includes events where their first datetime has not started yet.

filter_events_widget_example.php

<?php
//* Please do NOT include the opening php tag, except of course if you're starting with a blank file

add_filter( 'FHEE__EEW_Upcoming_Events__widget__where', 'my_custom_events_widget_where_params', 10, 3 );
function my_custom_events_widget_where_params( $where, $category, $show_expired ) {
  if ( ! $show_expired ) {
    $where['Datetime.DTT_EVT_start'] = array( '>=', EEM_Datetime::instance()->current_time_for_query( 'DTT_EVT_start' ) );
    $where['Datetime.DTT_order'] = 1;
  }
  return $where;
}