If you edit the Web part, you can edit the toolbar type to No Toolbar. Your New Item and Edit this list effectively disappears :)
If that removes your custom action, you need to hide the two buttons by using JQuery (or any other framework of your choice). The following code hides the entire toolbar, but you need to find the
class of each button to make the code workk for you.
$( document ).ready(function()
var test= $('tr').find('td.ms-list-addnew.ms-textXLarge.ms-list-addnew-aligntop.ms-soften');
Reference: How I can hide the new item/edit toolbar in SharePoint 2013 via JQuery