You can set socket to turning off when power consume is under x W for long, than y minutes, or so. Problem is not turning off, but turning on without loose comfort. This can be solved with solutions like Home Assistant (or basic solutions like Tapo/Kasa, Aqara, Tuya, ...), where you can set something like "when TV is turned on, turn on socket with amplifier too" or so, but using this solution only for amp is maybe overkill. Next solution is "master-slave socket strip". But, when you only want to start listening music, you don't have trigger to that, so, you must turn socket on before you start listening (and wait near minute to start Wiim). Turning socket on you can do in Smartphone app, or maybe with some solution, that turning socket based on remote controll (possible with Home Assistant too), or you can use "wire remote socket" with push button. But, when you start using something like Home Assistant (which now support Wiim too) and you install some "smart" solutions to your home, you have next nice options like "when I turn on TV, then turn on Wiim, cover windows, disable kitchen and bath-room fans, switch to "night light scene" and so
You can set to turning on Wiim while you sit on sofa long than 30seconds (you need presence sensor), or just when someone is presence in room, house, near house, .... Then you can turn off Wiim (not only) while nobody in house, room, on sofa, ...
But, I recommend simply to leave Wiim on and use native standby, if OP not use smart devices.
But, I recommend simply to leave Wiim on and use native standby, if OP not use smart devices.