Can the external power be supplied in one of the ports of the usb hub?
Some cheap usb hubs, are really really cheap, and do not follow appropriate standards. Some are so cheap, that they have all of four components. An all in one usb hub IC, a crystal clock, a decoupling capacitor, and an led. Input VCC is tied together to all the output VCC. These are not the best devices in the world.

That said, because VCC is tied together, you can have problems if you connect an external supply. You will have two competing supplies, and will put your computer at risk.

There are two solutions. The first, cut the VCC trace between the input usb port, and the output usb ports, but leave the usb hub ic connected to the input. By USB Standards, self-powered usb hubs and devices should still have the hub controller  connected to the host usb, at the very least for signaling purposes. This option makes it so you always have to have the external power connected for anything to work.

The second option is to still cut the trace, but add a diode to protect the input usb host. This will allow the hub to still be used without the external supply, and to be used with it as well. Be sure to pick a diode that can handle at least 1 Amp with a low forward voltage drop, like a schottky diode (0.2v typical), to minimize any issues with low voltage.

Picture of said extremely cheap usb hub. Diodes are a cheap way to bring 5v down to ~3.3v for the hub ic. No components on the other side:


I am not responsible if you break your usb hub, your computer, your external power supply, or any of your usb devices. Good Luck.