PHP Code (plugin or functions.php):

// Register the dashboard widget
function notes_dashboard_widget() {
    wp_add_dashboard_widget(
        'quick_notes_widget',
        'Quick Notes',
        'render_notes_widget'
    );
}
add_action('wp_dashboard_setup', 'notes_dashboard_widget');

// Render the widget content
function render_notes_widget() {
    // Get saved notes
    $notes = get_option('quick_notes', []);
    
    // Output the widget HTML
    ?>
    <div id="quick-notes-widget">
        <ul id="notes-list">
            <?php foreach ($notes as $index => $note): ?>
                <li>
                    <span class="note-text"><?php echo esc_html($note); ?></span>
                    <button class="delete-note" data-index="<?php echo $index; ?>">×</button>
                </li>
            <?php endforeach; ?>
        </ul>
        
        <div class="add-note-form">
            <textarea id="new-note" placeholder="Add a quick note..."></textarea>
            <button id="add-note-btn" class="button button-primary">Add Note</button>
            <span class="spinner"></span>
        </div>
    </div>
    <?php
}

// Enqueue the JavaScript
function enqueue_notes_widget_script() {
    // Only on dashboard page
    if (get_current_screen()->base !== 'dashboard') {
        return;
    }
    
    wp_enqueue_script(
        'notes-widget-js',
        get_template_directory_uri() . '/js/notes-widget.js',
        ['jquery'],
        '1.0',
        true
    );
    
    wp_localize_script('notes-widget-js', 'notesWidget', [
        'ajaxurl' => admin_url('admin-ajax.php'),
        'nonce' => wp_create_nonce('notes_widget_nonce')
    ]);
}
add_action('admin_enqueue_scripts', 'enqueue_notes_widget_script');

// AJAX handler for adding notes
function ajax_add_note() {
    check_ajax_referer('notes_widget_nonce', 'nonce');
    
    // Check permissions
    if (!current_user_can('edit_posts')) {
        wp_send_json_error('Permission denied');
    }
    
    $note = sanitize_textarea_field($_POST['note']);
    if (empty($note)) {
        wp_send_json_error('Note cannot be empty');
    }
    
    // Get existing notes and add the new one
    $notes = get_option('quick_notes', []);
    $notes[] = $note;
    
    // Save the updated notes
    update_option('quick_notes', $notes);
    
    // Return the index of the new note
    wp_send_json_success([
        'index' => count($notes) - 1,
        'note' => $note
    ]);
}
add_action('wp_ajax_add_note', 'ajax_add_note');

// AJAX handler for deleting notes
function ajax_delete_note() {
    check_ajax_referer('notes_widget_nonce', 'nonce');
    
    // Check permissions
    if (!current_user_can('edit_posts')) {
        wp_send_json_error('Permission denied');
    }
    
    $index = intval($_POST['index']);
    
    // Get existing notes and remove the specified one
    $notes = get_option('quick_notes', []);
    
    if (isset($notes[$index])) {
        unset($notes[$index]);
        $notes = array_values($notes); // Reindex the array
        update_option('quick_notes', $notes);
        wp_send_json_success();
    } else {
        wp_send_json_error('Note not found');
    }
}
add_action('wp_ajax_delete_note', 'ajax_delete_note');

JavaScript (notes-widget.js):

jQuery(document).ready(function($) {
    // Add a new note
    $("#add-note-btn").on("click", function() {
        let $button = $(this);
        let $spinner = $button.next(".spinner");
        let noteText = $("#new-note").val().trim();
        
        if (!noteText) {
            alert("Please enter a note");
            return;
        }
        
        // Disable button and show spinner
        $button.prop("disabled", true);
        $spinner.css("visibility", "visible");
        
        // Send AJAX request
        $.ajax({
            url: notesWidget.ajaxurl,
            type: "POST",
            data: {
                action: "add_note",
                nonce: notesWidget.nonce,
                note: noteText
            },
            success: function(response) {
                if (response.success) {
                    // Add the new note to the list
                    let newNote = $("<li>")
                        .append($("<span>").addClass("note-text").text(response.data.note))
                        .append($("<button>").addClass("delete-note").attr("data-index", response.data.index).text("×"));
                    
                    $("#notes-list").append(newNote);
                    
                    // Clear the textarea
                    $("#new-note").val("");
                } else {
                    alert("Error: " + response.data);
                }
            },
            error: function() {
                alert("An error occurred while saving the note");
            },
            complete: function() {
                // Re-enable button and hide spinner
                $button.prop("disabled", false);
                $spinner.css("visibility", "hidden");
            }
        });
    });
    
    // Delete a note (using event delegation)
    $("#notes-list").on("click", ".delete-note", function() {
        let $button = $(this);
        let $item = $button.closest("li");
        let index = $button.data("index");
        
        // Confirm deletion
        if (!confirm("Are you sure you want to delete this note?")) {
            return;
        }
        
        // Fade out the item
        $item.fadeOut(300);
        
        // Send AJAX request
        $.ajax({
            url: notesWidget.ajaxurl,
            type: "POST",
            data: {
                action: "delete_note",
                nonce: notesWidget.nonce,
                index: index
            },
            success: function(response) {
                if (response.success) {
                    // Remove the item from DOM completely after fade
                    $item.remove();
                } else {
                    // Show the item again if there was an error
                    $item.fadeIn(300);
                    alert("Error: " + response.data);
                }
            },
            error: function() {
                // Show the item again if there was an error
                $item.fadeIn(300);
                alert("An error occurred while deleting the note");
            }
        });
    });
});

CSS (Optional):

#quick-notes-widget {
    padding: 0 8px;
}

#notes-list {
    margin: 0 -8px;
    max-height: 200px;
    overflow-y: auto;
}

#notes-list li {
    display: flex;
    justify-content: space-between;
    align-items: center;
    padding: 8px;
    margin-bottom: 5px;
    background: #f9f9f9;
    border-left: 3px solid #0073aa;
}

#notes-list .note-text {
    flex: 1;
}

.delete-note {
    background: none;
    border: none;
    color: #a00;
    cursor: pointer;
    font-size: 18px;
    padding: 0 5px;
}

.delete-note:hover {
    color: #dc3232;
}

#new-note {
    width: 100%;
    margin: 10px 0;
    min-height: 60px;
}

.add-note-form {
    margin-top: 15px;
    position: relative;
}

.add-note-form .spinner {
    float: none;
    margin-top: 4px;
}