Here's a circuit from Colin Mitchell's Talking Electronics
website that can likely be changed to fit your needs.
In your case you would need to change the timing as well as how it is activated.
The timing is controlled by the resistors and capacitor on the left-hand side. In your case the capacitor will need to be much smaller and the pot can be removed if you don't need the timing to be adjustable.
The activation is controlled by the 'start' switch which can be replaced with a capacitor. You can leave the 'start' switch if you need to activate the relay manually. In that case just put the capacitor in parallel with it. You might need to experiment with the capacitor size to see what works best. The bigger, the better since it only needs to 'bootstrap' the timer long enough to turn the relay on.
The relay shown it the schematic will likely have to be a different relay from the relay that you are going to power for 10 seconds. That is, you will need two relays. The one shown in the schematic can be very small since it is just powering itself and the components to the left of it.
Because you will be driving two relays, you will need to be mindful about not exceeding the 555's maximum output current. Be careful about which 555 you use since they each have differences in the maximum output characteristics.
Refer to the 555 datasheet for additional details.
Edit: The more I think about it, I don't think this will work as I envisioned. It would likely turn right back on after it shutting itself off.