Yes, the alarm clock can be left permanently plugged in via USB (5V adapter not supplied).
The alarm clock is rechargeable (USB-C) and also runs on an internal battery, with a claimed autonomy of up to 3 months after full charge; the USB-C cable is supplied, but the 5V wall adapter is not.
I'm not entirely sure about the recommendations for prolonged use (e.g. leaving it constantly plugged in can, depending on the device, slightly reduce battery life) - the documentation supplied doesn't give precise instructions on this subject.