This turned out to be the Gamut File Viewer (
www.gamutsoftware.com), which I have just started using (that should have been my clue). Reading the docs, it seems it has the functionality to clear the log file, which probably means it keeps a permanent lock on the file.
I am surprised that I was even denied Read permission though.
I was going to delete this question, but then I thought it might have value to anyone else who chooses to use this fairly common set of tools.