I get this in my debug log every so often. Is there a way to log the IP address of the request causing it?
NOTICE: Use of undefined constant ADMIN_RIGHTS - assumed 'ADMIN_RIGHTS' in /(path)/zenphoto/zp-core/reconfigure.php on line 110 checkSignature called from reconfigureAction (reconfigure.php [12]) from require_once (functions-basic.php [161]) from require_once (functions.php [17]) from include (index.php [14]) from index.php [32]
I did not cause it. I tried to correlate with the raw access logs but I couldn't find anything that stood out. The MySQL max connections was also triggered while these attempts were being made.

Suppose I can edit the reconfigure.php file and log all accesses to it.

reconfigure.php is run automatically when Zenphoto checks for changes to the install (security check of files not in the catalog or time changes) and if the server changes (Apache, PHP, MySQL version changes etc.). You see this - if loggedin as an admin - when Zenphoto reports it may have found changes and suggest to re-run setup.

Why the constant ADMIN_RIGHTS is not defined, I cannot answer at the moment.

Well, I added a log to that file. I trigger that section when loading admin.php (overview tab). That notice doesn't appear in the log though when I am accessing it.

Just wait and see until it happens again.

That check runs in the background actually but of course only puts out a note if something has changed.