Please do the following:
Launch System Preference > Security and Privacy > General tab. In Allow apps downloaded from the: section, which option is selected? If Mac App Store and identified developers is not selected, please select this option.
It was selected.
I have now succeeded in completing the update. I decided the OS was getting tired, so I closed down, gave it half a minute rest and booted up from scratch again. Then, before loading any but the start-up programs, and before Safari, I loaded the dmg, clicked the installer, and all went as usual. So, if you come across that again, suggest the client does a complete power-off and reboot.
Another thing you may want to take note of is that every time I have done an update I clicked on the 'Allow Adobe to install updates' button, as prompted in the preliminary stages of the update. When you prompted me to go to System Preferences I checked the Flash Player preferences and found that it had the 'Notify me to install updates' button selected. I've now changed that to 'Allow Adobe to install updates'. I hope that now updates will install without me having to help them! You may want to amend the text at that preliminary stage to suggest people check their SysPrefs setting is also set to the recommended selection.
Thanks for your help. It may not look as if you provided the solution, but your input helped me to find the solution.