I have 2 laptops running LM 19.1 cinnamon version. i use an external drive to save home files such as downloads mint documents etc.
Since i tried to use virtual box in cinnamon it did not work to run windows 10, i now considering installing Linux mint mate on one of the laptops, my query is that will mate able to acres the external hard drive used in linux cinnamon
you could try a live usb to see how the external drive acts with mint mate on your system.
One test is worth a hundred opinions. 
But you did not say what was the file type / format of the external disk drives.
I found this page on the web that shows “most common file systems in use with the Linux world.”
https://help.ubuntu.com/community/LinuxFilesystemsExplained
I have found that the difference is how a specific file manager displays unmounted drives. Nautilus shows them before they are mounted while Caja does not, for example. Once it is mounted, it will be displayed.