These little mini-amps with the external supplies (and even some with internal supplies) are not intended to be turned off and started back up repeatedly. The power supply should be plugged into a live outlet and left on all the time. The amp itself is also always on, but when "off" from it's power switch it is in a deep standby mode.
The reason for this is the large reservoir "Bulk" capacitors used on the amplifier boards and, often, in the power supply as well. When you first connect the Power cord to the supply there is a rather large inrush of current until it gets charged up and working. When you first connect the amp to the power supply there is another large inrush of current while the amp's big caps charge up and settle down. When you power cycle with both devices connected that inrush can be big enough to do damage.
Neither device is likely to be designed to handle this repeatedly. Each repetition puts a very large strain on the rectifier diodes and AC filters in the power supply. Each time causes some trivial amount of damage, until one day it starts acting up and eventually dies.
This issue came to the fore with people shutting down their PCs and then turning off the "babysitter switch" on the back of the case. We had a large rash of failed power supplies because the current inrush was popping the rectifiers inside. We ended up replacing quite a few under warranty and advising that the power switch be left on all the time. After that, very few supply failures.
So .... best advice... to avoid the "inrush" problem, hook up your mini-amp, plug it into a live outlet... and leave it "on" all the time. When not in use, use the "on-off" switch on the amplifier to put it into standby mode.