In order to help you I need to know exactly what plugin adds that menu items. Send me download link from wordpress.org or .zip copy (to support email) if it’s a paid product.
I use provided paid products for the testing purpose and at my local desktop only.

Contact Form 7 – “Integration” menu item is protected by ‘manage_options’ capability. If role has this capability it will be shown at ‘Admin Menu’ dialog for that role:
When I block it, it becomes unavailable for user with that role:
CF7 menu without Integration menu item