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HTML

<html>

  <head>
    <meta charset=utf-8 />
    <title>Reverse geocoding</title>
    <meta name='viewport' content='initial-scale=1,maximum-scale=1,user-scalable=no' />

    <!-- Load Leaflet from CDN -->
    <link rel="stylesheet" href="https://unpkg.com/leaflet@1.4.0/dist/leaflet.css" integrity="sha512-puBpdR0798OZvTTbP4A8Ix/l+A4dHDD0DGqYW6RQ+9jxkRFclaxxQb/SJAWZfWAkuyeQUytO7+7N4QKrDh+drA==" crossorigin="" />
    <script src="https://unpkg.com/leaflet@1.4.0/dist/leaflet.js" integrity="sha512-QVftwZFqvtRNi0ZyCtsznlKSWOStnDORoefr1enyq5mVL4tmKB3S/EnC3rRJcxCPavG10IcrVGSmPh6Qw5lwrg==" crossorigin=""></script>


    <!-- Load Esri Leaflet from CDN -->
    <script src="https://unpkg.com/esri-leaflet@2.2.3/dist/esri-leaflet.js" integrity="sha512-YZ6b5bXRVwipfqul5krehD9qlbJzc6KOGXYsDjU9HHXW2gK57xmWl2gU6nAegiErAqFXhygKIsWPKbjLPXVb2g==" crossorigin=""></script>


    <!-- Load Esri Leaflet Geocoder from CDN -->
    <link rel="stylesheet" href="https://unpkg.com/esri-leaflet-geocoder@2.2.13/dist/esri-leaflet-geocoder.css" integrity="sha512-v5YmWLm8KqAAmg5808pETiccEohtt8rPVMGQ1jA6jqkWVydV5Cuz3nJ9fQ7ittSxvuqsvI9RSGfVoKPaAJZ/AQ==" crossorigin="">
    <script src="https://unpkg.com/esri-leaflet-geocoder@2.2.13/dist/esri-leaflet-geocoder.js" integrity="sha512-zdT4Pc2tIrc6uoYly2Wp8jh6EPEWaveqqD3sT0lf5yei19BC1WulGuh5CesB0ldBKZieKGD7Qyf/G0jdSe016A==" crossorigin=""></script>

    <style>
      body {
        margin: 0;
        padding: 0;
      }

      #map {
        position: absolute;
        top: 0;
        bottom: 0;
        right: 0;
        left: 0;
      }

    </style>
  </head>

  <body>

    <div id="map"></div>



  </body>

</html>

JavaScript

var map = L.map('map').setView([40.725, -73.985], 7);

L.tileLayer('https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png', {
  attribution: '&copy; <a href="http://osm.org/copyright">OpenStreetMap</a> contributors'
}).addTo(map);

var geocodeService = L.esri.Geocoding.geocodeService();

map.on('click', function(e) {
  geocodeService.reverse().latlng(e.latlng).run(function(error, result) {
    L.marker(result.latlng).addTo(map).bindPopup(result.address.Match_addr).openPopup();
  });
});

var message;

geocodeService.reverse().latlng([40.725, -73.985]).run(function(error, result) {
  //alert(result.address.Match_addr); //this alert works here ok and can retur addrress
  message =result.address.Match_addr;
  ShowMsgAfterRequest();
})
function ShowMsgAfterRequest()
{
	alert(message)
}
//If you want to use the message just make sure the run function is already executed:
function checkMessageAfterSometime(time)
{
	setTimeout(function(){
  	alert(message)
  },time);
}
checkMessageAfterSometime(2000) //2 seconds