Please do not use get_template_part() in the Admin / Backend. It has been designed for template use (displaying the theme, please see codex), so this is intended to be used for templates and template-parts within themes:
Function Reference/get template part
Load a template part into a template
(other than header, sidebar, footer).
Makes it easy for a theme to reuse
sections of code in an easy to
overload way for child themes.
The function does actually includes general source code as well, but what you experience (as John pointed out) might be related to scope. As I do not know your code, I can not say for sure, if that is the exact cause of the issue you're facing (variable scope), so I tend to keep it simple in my answer:
The function is just not desgined to replace as standard php code include, for example to write an admin page.
So just do not use it if for your admin page unless you really know what you are doing.
If you want to load a php file into the global scope while looking in the child and parent theme directory, you can make use of a function that does the search for the file for you: locate_template():
Retrieve the name of the highest priority template file that exists.
Searches in the STYLESHEETPATH before TEMPLATEPATH so that themes which inherit from a parent theme can just overload one file.
For your given example include
include(TEMPLATEPATH."sitewideSettings.php"); this can be written as:
This allows you to control where and how the file is included (include/_once; require/_cone) and it prevents you to have the scope issue in your example which breaks the NONCE mechanism.
The "Please try again" message is there because wp-admin is not able to verify the request, more specifically the so called nonce. There can be two reasons for that: There is none in the request (e.g. the form does not provide any or it does not get's transferred or read by the admin code) or the nonce is invalid. Nonces always represent a chain of actions, and if that chain is broken, you'll get that message. But the message does not tells you where that chain is broken.
Those error messages in the admin are not always very informative and can be even misleading (Related: #14060). To find out what happens here, it's most effective to debug that on the system where the error appears as it can be quite specific which part of the chain is broken.
Please see the codex on how to create Admin Pages for your Theme or Plugin for more in-depth information about nonces, admin pages etc. . It's not possible for the Admin request to validate the nonce for some reason.