Custom Widget to list WordPress categories along with the Post count. Requires styling with CSS to blend it well with theme.

theme-widget-categories.php

<?php

// Adds widget: Categories
class RutuCategories_Widget extends WP_Widget {

    function __construct() {
        parent::__construct(
            'rutu_categories_widget',
            esc_html__( 'Rutu Categories', 'mfl' ),
            array( 'description' => esc_html__( 'Custom Categories widget.', 'mfl' ), ) // Args
        );
    }

    private $widget_fields = array(
        array(
            'label' => 'IDs of Categories to exclude',
            'id' => 'cw_exclude',
            'default' => '1',
            'type' => 'text',
        ),
    );

    public function widget( $args, $instance ) {
        echo $args['before_widget'];

        if ( ! empty( $instance['title'] ) ) {
            echo $args['before_title'] . apply_filters( 'widget_title', $instance['title'] ) . $args['after_title'];
        }

        $terms = get_terms( array(
            'taxonomy' => 'category',
            'hide_empty' => false,
            'childless' => true,
            'orderby' => 'name',
            'order' => 'ASC',
            'hierarchical' => false,
            'exclude' => $instance['cw_exclude'],
        ) );

        echo '<ul class="categories">';

        foreach($terms as $category) {
            echo '<li><a href="'. get_term_link($category->term_id) . '">'. $category->name . ' <span>('. $category->count . ')</span></a></li>';
        }
        echo '</ul>';

        echo $args['after_widget'];
    }

    public function field_generator( $instance ) {
        $output = '';
        foreach ( $this->widget_fields as $widget_field ) {
            $default = '';
            if ( isset($widget_field['default']) ) {
                $default = $widget_field['default'];
            }
            $widget_value = ! empty( $instance[$widget_field['id']] ) ? $instance[$widget_field['id']] : esc_html__( $default, 'mfl' );
            switch ( $widget_field['type'] ) {
                default:
                    $output .= '<p>';
                    $output .= '<label for="'.esc_attr( $this->get_field_id( $widget_field['id'] ) ).'">'.esc_attr( $widget_field['label'], 'mfl' ).':</label> ';
                    $output .= '<input class="widefat" id="'.esc_attr( $this->get_field_id( $widget_field['id'] ) ).'" name="'.esc_attr( $this->get_field_name( $widget_field['id'] ) ).'" type="'.$widget_field['type'].'" value="'.esc_attr( $widget_value ).'">';
                    $output .= '</p>';
            }
        }
        echo $output;
    }

    public function form( $instance ) {
        $title = ! empty( $instance['title'] ) ? $instance['title'] : esc_html__( '', 'mfl' );
        ?>
        <p>
            <label for="<?php echo esc_attr( $this->get_field_id( 'title' ) ); ?>"><?php esc_attr_e( 'Title:', 'mfl' ); ?></label>
            <input class="widefat" id="<?php echo esc_attr( $this->get_field_id( 'title' ) ); ?>" name="<?php echo esc_attr( $this->get_field_name( 'title' ) ); ?>" type="text" value="<?php echo esc_attr( $title ); ?>">
        </p>
        <?php
        $this->field_generator( $instance );
    }

    public function update( $new_instance, $old_instance ) {
        $instance = array();
        $instance['title'] = ( ! empty( $new_instance['title'] ) ) ? strip_tags( $new_instance['title'] ) : '';
        foreach ( $this->widget_fields as $widget_field ) {
            switch ( $widget_field['type'] ) {
                default:
                    $instance[$widget_field['id']] = ( ! empty( $new_instance[$widget_field['id']] ) ) ? strip_tags( $new_instance[$widget_field['id']] ) : '';
            }
        }
        return $instance;
    }
}

function register_rutu_categories_widget() {
    register_widget( 'RutuCategories_Widget' );
}
add_action( 'widgets_init', 'register_rutu_categories_widget' );