It appears your Home directory permissions are incorrect. yay isn’t able to write to it’s cache directory: ~/.cache/yay/.
No wonder if you’re running commands with sudo that you should not be.
See:
https://serverfault.com/questions/35076/need-to-fix-file-permissions-in-a-users-home-directory
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Please edit your post accordingly. The Trizen output is done properly, the Yay output is not.