Getting back to OP's question; you should be able to install Cinnamon as an additional desktop environment on Linux Mint MATE, but it may not give you as fully integrated or as suave an experience as installing Linux Mint Cinnamon. If you want to add Cinnamon to Linux Mint MATE, install the cinnamon, cinnamon-themes, and mint-artwork-cinnamon packages. Recommend you use apt for that from the terminal, and you save the output so you can ask for support here if anything doesn't work quite right yet. You can copy text to/from the terminal, see its Edit menu.
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apt install cinnamon cinnamon-themes mint-artwork-cinnamon
After installation, you should be able to choose to run a Cinnamon session from the login screen (see controls at the bottom).
You could also install Linux Mint Cinnamon alongside your current Linux Mint MATE installation, in its own partition. This way, if you want to continue with only MATE or Cinnamon at some future date, you can just remove either the one or the other partition and increase the remaining partition to absorb the freed up disk space.