Custom search results page not working with empty search

I’m not overly familiar with search results pages and how you edit them, but a quick fix for the problem you have is to check for an empty string in the first bit of code and say “Please enter a search term” or something if there isn’t one entered. You’d achieve that with something like:

<?php
  if (empty($s)) {
     _e('Please enter a search term');
  } else {
    $allsearch = new WP_Query("s=$s&showposts=-1");
    $key = esc_html($s, 1);
    $count = $allsearch->post_count; 
    _e(''); _e('&#8220;'); _e('<span class="searchTerm">');
    echo $key; 
    _e('</span>'); _e('&#8221;'); _e(' - ');
    echo $count . ' '; _e('found');
    wp_reset_query(); 
  }
?>

Extra stuff not relevant to your question: For the second bit of code it may be worth looking at the docs here https://developer.wordpress.org/reference/functions/query_posts/ and switching to using get_posts, but as I said I’m not sure if using query_posts might be ok in this case for search pages. Certainly the docs say it shouldn’t be used in your own themes.