The answer popped up right away when I put "/tmp does not exist or is not writable" in Google.
The only thing that can cause this is a read-write permissions issue with tmp. This is not a Stackposts issue, but a system configuration issue on your end, so if you do not contro9l it, I would get with your hosting provider.
With extremely little to go off of, my guess is user "admin" (owner and group owner of your tmp dir) is not actually the user who should be owning the tmp? Admin and admin group own it, so if your regular, everyday user is not admin (not the user account logged into when installing Stackpost), then chown commands needs to be done by the admin account, giving ownership and group ownership to the user account who initially installed Stackpost. So, I do not know what level of access you have to this system, but user admin or root will need to run the following command.
chown owner-user:owner-group /path/to/tmp/dir
Make sure to change the "owner-user" in above command to the user you log in with (normaluser1:normaluser1).
For permissions of the tmp directory, it's safe to say 755 is good, with files being 644, I would check the perms also.