Hi nauna,

You can use JavaScript setTimeout and clearTimeout function.

Refer below JavaScript code.

<form id="form1" runat="server">
    <div>
        <ul id="menu"></ul>
    </div>
</form>
<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript" src="https://ajax.aspnetcdn.com/ajax/jquery.ui/1.8.9/jquery-ui.js"></script>
<link rel="stylesheet" type="text/css" href="https://ajax.aspnetcdn.com/ajax/jquery.ui/1.8.9/themes/blitzer/jquery-ui.css" />
<script type="text/javascript">
    $(function () {
        $('ul').menu();
        $.ajax({
            url: "Handler.ashx",
            method: 'GET',
            dataType: 'json',
            contentType: "application/json; charset=utf-8",
            success: function (data) {
                $.each(data, function () {
                    var li = $('<li value="' + this.Id + '">' + this.MenuText + '</li>');
                    if (!this.Active) {
                        li.addClass('ui-state-disabled');
                    }
                    li.appendTo($('#menu'));
                });
            }
        });

        var timer;
        $('body').on('mouseenter', 'ul li', function () {
            var menu = $(this);
            var id = $(this).attr('value');
            timer = setTimeout(function () {
                $.ajax({
                    url: "Handler.ashx?ParentId=" + id,
                    method: 'GET',
                    dataType: 'json',
                    contentType: "application/json; charset=utf-8",
                    success: function (data) {
                        var ul = $('<ul></ul>').empty();
                        $.each(data, function () {
                            var li = $('<li value="' + this.Id + '">' + this.MenuText + '</li>');
                            if (!this.Active) {
                                li.addClass('ui-state-disabled');
                            }
                            li.appendTo($(ul));
                        });
                        $(ul).appendTo($(menu));
                    }
                });
            }, 500);
        });

        $('body').on('mouseleave', 'ul li', function () {
            $(this).find('ul').remove();
            clearTimeout(timer);
        });
    });
</script>