HTML Code

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="utf-8">
  <title>one click mouseenter demo</title>
  <style>
    body { font-family: sans-serif; padding: 16px; }
    .target {
      display: inline-block; padding: 20px 28px; background: #dbeafe;
      border: 2px solid #3b82f6; border-radius: 8px; cursor: pointer; font-weight: bold;
    }
    .target:hover { background: #bfdbfe; }
    .count { font-size: 2rem; color: #dc2626; margin: 12px 0; font-weight: bold; }
    #log { color: #374151; max-width: 400px; }
  </style>
</head>
<body>
  <div class="count">0</div>
  <div class="target">Hover/click me</div>
  <div id="log">Count increases on first mouseenter and first click (max 2).</div>
  <script src="https://code.jquery.com/jquery-3.7.1.min.js"></script>
  <script>
    var n = 0;
    $(".target").one("click mouseenter", function(event) {
      $(".count").html(++n);
      $("#log").text("The " + event.type + " event happened! Count = " + n);
    });
  </script>
</body>
</html>

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jQuery .one('click mouseenter') — once per event type (Try It)