The fix to this issue is because the form key validation is missing on check out. You can view the fix here How to fix Magento checkout after upgrading
Blog // Web Hosting // How to fix Magento checkout redirecting to cart after upgrading to 1.9.3.3 or 1.9.3.4
The fix to this issue is because the form key validation is missing on check out. You can view the fix here How to fix Magento checkout after upgrading