there was a sharp SNAP sound - like a fuse or a circuit blowing - and then a puff of smoke from the back of the machine. And quite a nasty smell, so I unplugged it quickly.
That's often the X2 safety capacitor that does that. The good news is that it's a simple fix to replace it; it's usually located underneath the IEC inlet connector, surrounded by heat shrink. Technically the power supply doesn't even need that capacitor to operate, its job is to reduce high frequency switching noise that would be otherwise be conducted back into the AC mains supply. All switchmode power supplies need a similar form of high frequency AC decoupling to meet conducted emissions standards.
That said, I'd recommend you get it looked at by someone competent working on AC power supplies as it contains dangerous internal working voltages. I can do the job if you get stuck, though it's not that economic to send a power supply to New Zealand for a simple repair job.
http://amiga.serveftp.net
