Move closing </a> tag to the end of a containing element

Avoid messing with html directly, it's better not to change it or overwrite. All you need to do is to take next text sibling Node and append to previous a:

$('.text a').each(function() {
  $(this).append(this.nextSibling)
})

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

<div class="text">
  <a href="link">text</a> and more text
</div>

If necessary you can check for the next node to be TextNode, if you need to skip element nodes:

if (this.nextSibling.nodeType === 3) {
  $(this).append(this.nextSibling)
}

You need to use .append( function ) to insert nextSibling of anchor into it.

$(".text a").append(function(){
  return this.nextSibling
});

$(".text a").append(function(){
  return this.nextSibling
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div class="text">
  <a href="link">text</a> and more text
</div>

Also you can use .html( function ) instead and then remove next sibling using .remove()

$(".text a").html(function(i, h){
  return h + this.nextSibling.nodeValue;
})[0].nextSibling.remove();

Or in one line using ES6

$(".text a").html((i,h) => h+this.nextSibling.nodeValue)[0].nextSibling.remove();

$(".text a").html(function(i, h){
  return h + this.nextSibling.nodeValue;
})[0].nextSibling.remove();

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div class="text">
  <a href="link">text</a> and more text
</div>

Or using pure javascript

var ele = document.querySelector(".text a");
ele.innerHTML += ele.nextSibling.nodeValue;
ele.nextSibling.remove();

var ele = document.querySelector(".text a");
ele.innerHTML += ele.nextSibling.nodeValue;
ele.nextSibling.remove();

<div class="text">
  <a href="link">text</a> and more text
</div>

HTML/JavaScript doesn't work in a way that you can "move" a closing tag like that, but what you can do is move the text. Also, you don't need jQuery to do it; it's very easy to do with vanilla JavaScript:

let link = document.querySelector('.text a')
let textAfterLink = link.nextSibling
link.appendChild(textAfterLink)

<div class="text">
  <a href="link">text</a> and more text
</div>

You can first get the HTML inside the div with class text and then replace the closing tag </a> with '' then finally append a closing </a> tag to the replaced string so that you get what you expect:

var aHTML = $('.text').html();
aHTML = aHTML.trim().replace(/<\/a>/, '') + '</a>';
$('.text').html(aHTML);

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div class="text">
    <a href="link">text</a> and more text
 </div>

Here you have one approach that will find all unwraped text inside the element with class .text and append all of these texts to the first <a> child. This approach uses the content() method chained with a filter() using the addequated condition for remove the texts children, while at the same time they are appended to the <a> element.

$('.text').each(function()
{
    $(this).contents().filter(function()
    {
        // Filter text type only.
        return (this.nodeType === 3);
    })
    .appendTo($(this).find("a:first-child"));
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

<div class="text">
    <a href="link">text</a> and more text
</div>

<hr>

<div class="text">
  <a href="link">Rise up this morning</a>
  Smile with the rising sun
  <br>
  Three little birds
  <br>
  Pitched by my doorstep
  <br>
  <p>DON'T TOUCH THIS ONE!</p>
  Singing sweet songs
  <br>
  Of melodies pure and true
  <br>
  Sayin': This is my message to you
  <br>
  Saying, don't worry about a thing
  <br>
  'Cause every little thing
  <br>
  Gonna be all right
</div>