Gives an example of how to allow editor role access to use siteimprove plugin.

wpse318816-editor-access.php

<?php
/**
 * Plugin Name: WPSE318816 Editor Access
 * Description: Gives an example of how to allow editor role access to use siteimprove plugin. See https://wordpress.stackexchange.com/questions/318816/adding-menu-item-to-wordpress-admin-bar-for-editors-to-edit-one-plugins-setting.
 * Plugin URI: https://gist.github.com/timelsass/87b24de86a9cce6633bbc55ed4213a5f
 * Author: Tim Elsass
 * Author URI: tim.ph
 */

// Checks if the current user is an editor role.
function wpse_318816_get_current_user_role() {
	if ( is_user_logged_in() ) {
		$user = wp_get_current_user();
		return in_array( 'editor', $user->roles );
	}
}

// Adds the siteimprove menu for the editor role's capability.
function wpse_318816_add_editor_menu() {
	if ( wpse_318816_get_current_user_role() ) {
		add_menu_page(
			'Siteimprove Plugin',
			'Siteimprove',
			'edit_others_pages',
			'siteimprove',
			'Siteimprove_Admin_Settings::siteimprove_settings_form'
		);
	}
}

add_action( 'admin_init', 'wpse_318816_add_editor_menu' );

// Remaps edit capability to the user role.
function wpse_318816_add_editor_cap( $caps = array(), $cap = '', $user_id = 0, $args = array() ) {
	global $pagenow;

	if ( $cap !== 'manage_options' ) {
		return $caps;
	}

	// Remove the filter to prevent resursive loops.
	remove_filter( 'map_meta_cap', 'wpse_318816_add_editor_cap', 11 );

	// Abort for admin.
	if ( current_user_can( 'administrator' ) ) {
		return $caps;
	}

	// Only do the following stuff if the user is the admin.
	if ( wpse_318816_get_current_user_role() ) {

		// Check if you're on the adminpage for siteimprove.
		if ( ( $pagenow === 'admin.php' && $_GET['page'] === 'siteimprove' ) ||

			// Check if you're submitting the token request form.
			( $pagenow === 'options.php' && $_POST['option_page'] === 'siteimprove' ) ) {

				// Set caps to edit_others_pages for the editor role.
				$caps = array( 'edit_others_pages' );

				// Re-add the filter.
				add_filter( 'map_meta_cap', 'wpse_318816_add_editor_cap', 11, 4 );

			// Remove the menu page as we are remapping manage_options to edit_others_pages.
			remove_menu_page( 'siteimprove' );
		}
	}

	return $caps;
}

add_filter( 'map_meta_cap', 'wpse_318816_add_editor_cap', 11, 4 );