Thanks for that suggestion deHakkelaar.
I did "sudo nmap --script broadcast-dhcp-discover" to find out exactly what information the ASUS Router's DHCP Service was handing out. The following is an extract of the output of that nmap command:
"Domain Name Server: 192.168.2.22, 192.168.2.1"
So, the ASUS Router's DHCP Service is handing out an unwanted secondary DNS address, even though the Pi-Hole address is the only DNS service configured.. My Pi-Hole Server is at 192.168.2.22, and the ASUS Router is at 192.68.2.1.
There doesn't seem to be anyway to disable the ASUS Router's own DNS in the Router's web interface. I've been planning to change the ASUS Router over to DD-WRT to enable a few other additional features, so will install DD-WRT hoping that I will then get enough additional configuration flexibility in DD-WRT to enable me to disable the router's DNS Service --- or perhaps to install Pi-Hole on DD-WRT - but that idea needs more research - I'm not sure if it's possible yet.
So, thanks again for your suggestion.
Regards,
RoyG