To protect against power cuts, I bought a UPS.
No mention is made in the instructions about whether, when I switch off the computer, I should switch of the power supply or leave it on.
I have no idea how these devices work so I don't know if they need to be left on.
There's no benefit in leaving it switched on and it will use a small amount of power (you'll probably find that the casing is slightly warm), so best to switch off.
FYI, these units usually use a 12V sealed lead acid battery (similar to those used as battery back-ups in burglar alarm systems). The batteries are readily available (e.g. TLC Direct) and may need replacing every 5 years or so).
Wasn't there a post about this a long time ago? Something about the battery. Someone said it should be continuously trickle fed to keep it fully charged. If you keep discharging and charging it reduces the life of the battery. I've always left it on.