if you look into the podpress_class.php file you will see that the function checkWritableTempFileDir() checks whether this podpress_temp folder is in place and writeable.
The error occurs in line 647 where it tries to create the folder which name should be stored in the variable $this->tempFileSystemPath. Usually this variable is not empty and by default just an absolute path which ends with .../wp-content/uploads/podpress_temp.
Maybe the cause of this error message is because of a permissions problem.
(The PHP manual on mkdir says [...]When safe mode is enabled, PHP checks whether the directory in which the script is operating has the same UID (owner) as the script that is being executed.[...])
Or maybe it is a value which mkdir cannot handle.
It would be great if you could find out more about the content of this $this->tempFileSystemPath variable.
I have written a little WP plugin for dumping the content of a PHP variable into a log file. It is called printphpnotices. You can download it from my website.
Install it like other plugins, activate it and insert right after line 619 of the file podpress_class.php the call:
After that update a post and look into the folder of this little debug plugin (/wp-content/plugins/aaaprintphpnotices). There should be a file called printphpnotices_log.dat which should contain the content of this variable.
Besides this you could suppress the error message(s) if you would put an @ in front of the mkdir command.
$mkdir = @mkdir($this->tempFileSystemPath);
That should be ok because the function will return a custom error message if necessary or at least give back a matching result.
But before you try this, try to retrieve the content of this variable.