Telling me that multiple DHCP doesn’t have any effect on your problems and telling me that you cannot access your My Cloud is like telling me that your brakes has nothing to do with your gas pedal.
There are two basic component to a router that makes it a router and that is the DHCP and DNS. These two determines the IP or the address of the device and redirects anybody asking for that address to that device. The DNS IP within your local network usually points back to your router which is usually 192.168.0.1 and your router determines whether the ip that you are requesting is inside or outside your local network. So if your My Cloud is connected to a second router that has its own DHCP through the WAN port of that router, when you request access to the My Cloud, you get routed to the second router IP but no further.
When your device like a My Cloud boots up, it will send out a signal asking for an IP and if you have multiple DHCP, everyone will send out an IP and at this point, I have no idea which IP your My Cloud will take. According to your description, It seems that it would take the IP from your Wifi router thus you would have access to the My Cloud via Wifi. If you connect your gigabit ethernet directly to your Wifi router, you probably would be able to access the My Cloud just as you would via Wifi.
Having multiple DHCP on a network confuses all the routers to where your individual devices are. It may appear to work mostly but as you have found out, switching to ethernet hides your My Cloud from you because the path to your My Cloud is not available to your connection point.
Disconnecting various routers from the mixed up Network, thus removing the multiple DHCP, may seem to have no effect and that is because every device you have is already confused and requires a re-boot starting with the initial router followed by each device to re-acquire a new IP from one DHCP source.
Ok fine having multiple routers with DHCP can work if you connect from one of the LAN port of a router to the WAN port of another router, those becomes two independent network and the two DHCP won’t affect each other but in order to access the my cloud on the second network you need to use the assigned IP from the second network and the subnet must be the same (maybe). You won’t be able to browse for your My Cloud because your DNS won’t have any knowledge of where your My Cloud is. I do know that a static IP on your My Cloud (outside the DHCP range) will definitely allow you to access the My Cloud across two different networks (same subnet).
If you wish to have wifi, you can set up your main Wifi router as your main router. Turn off DHCP on your ISP router (ISP = internet Service Provider/Main router) and if you can there might be an option to turn on Bridge mode, meaning that your ISP router will become just an internet bridge connector to your Wifi router.
Optionally you could do what I did is to have the Wifi Router operate as a bridge router with the IPS router running DHCP and all the Wifi Router does is provide Wifi connection (DHCP off).
I hope all this info helps you figure it all out and again good luck.