Moving on after "password and re-typed password" verification

I think to resolve this issue what you can do is initially hide the anchor tag and dynamically trigger it based on your verification process success or failure like this:

Html:

<!-- Add a placeholder element for the statement -->
<div id="next-stage" style="display: none;">
  <a href="mainmenu.html">Move on to the next stage</a>
</div>

<!-- Add a button to trigger the password verification -->
<button onclick="checkPassword()">Submit</button>

JavaScript:
On the function side of things given that you already have the logic to verify the integrity of the password, you can do something like this:

Note: Just keep in mind that the password verification is hardcoded by me to test the logic in the below snippet, u might need to write your own logic based on the use case.

<!-- begin snippet: js hide: false console: true babel: false -->

<!-- language: lang-js -->

function checkPassword() {
  // Perform the password verification
  var isPasswordValid =true // Your password verification logic here

  if (isPasswordValid) {
    // Show the next stage statement
    document.getElementById("next-stage").style.display = "block";
    window.alert("Password verification successful!");
  } else {
    window.alert("Password verification failed!");
  }
}

<!-- language: lang-html -->

<!-- Add a placeholder element for the statement -->
<div id="next-stage" style="display: none;">
  <a href="mainmenu.html">Move on to the next stage</a>
</div>

<!-- Add a button to trigger the password verification -->
<button onclick="checkPassword()">Submit</button>

<!-- end snippet -->

I'm sorry, but I don't quite understand the question.

Maybe you are looking for something like this.

<!-- begin snippet: js hide: true console: true babel: false -->

<!-- language: lang-js -->

$(document).ready(function(){
    $('#btn').attr('disabled',true);
    $('#rpw').keyup(function(){
        if($(this).val().length !=0)
          $('#btn').attr('disabled', false);            
        else
          $('#btn').attr('disabled',true);
    })
    $("#btn").click(function(){
      $("#pw").val() == $("#rpw").val() ?
      (window.alert('Successful!'), 
      $("#moveToNext").css("display", "block"))
      : (window.alert('Password not match!'),
      $("#moveToNext").css("display", "none"))
  });
});

<!-- language: lang-html -->

<script src="https://code.jquery.com/jquery-3.6.0.min.js"></script>

<label>Password: </label><input id="pw" type="password">
<br>
<label>Re-type Password: </label><input id="rpw" type="password">
<br>
<button id="btn">Check</button>
<div style="display:none;" id="moveToNext">
  <a href="mainmenu.html" onClick="checkPassword()">Move on the next stage</a>
</div>

<!-- end snippet -->