How can I show the contents of a file at a specific state of a git repo?

It’s probably problem with your path specification.

This works, shows version of Makefile in commit b1b22df407417...

git show b1b22df407417:Makefile

Or current version in master branch

git show master:Makefile

Or current version in exper branch:

git show exper:Makefile

Or previous version on the exper branch:

git show exper^:Makefile

And so on