Yes, mine does and actually will do it longer if there is more load on the motor such as with a Super Platter. I think it's a soft start clutch on the motor.
I had the same situation on a Scout (with the 300rpm motor upgrade) and also on the new Aires III that I replaced the Scout with. I thought it sounded like "belt slap". Like the belt would hold back just a bit on the initial load but then kind of "snap" when it got up to speed. I have a Super Platter arriving today or tomorrow for the Aires. I'll see if it is more delayed as Mike described.
I, too, have the same clicking sound on my Scoutmaster. At first I thought it was the belt. Concurrent with turning on the motor, I also help lesson the start-up load by giving the record platter a little twist. But I have also had the same noise since day one. But not one complaint on performance with the Scoutmaster. I love it. I don't pay attention to the noise. It goes away after the table is at speed, usually about 1 second or so.