When budgies are moulting, they can feel more itchy from the growing feathers and you will also notice an increase on preening. During a preen, a few feathers can naturally fall off, especially when moulting.
Plucking feathers is a whole different business, this is a violent action where the budgie will intentionally pluck the feathers out.
To help with the general discomfort and itch from the moult, you can offer Anzu a means for him to bathe, by placing a shallow dish of water.
You can also gently mist your budgie with a spray bottle, if he's not keen on taking baths.
Supplementing your budgie's diet with egg food will also be vital as this will help in replenishing the energy lost during the moult.
If you notice a very abrupt change on your Anzu's activity (sleepy and puffed up all day) and if he is spending time at the bottom of the cage, then the best thing to do is to book your budgie an appointment with an avian vet specialist. The moult could have brought up an underlying illness or made him more susceptible to fall ill.
I hope your Anzu feels better soon.