I own a bravo myself and I fiddled with the air filter box myself recently, so I wanted to share my experiences.
What I think happened is the following: the person you bought your bravo from changed the jet size (put in a higher number jet) in the carburetor with a result that more air was needed than the air box would supply, so he probably took that off. (with the possibility of dust/mud/water getting sucked into your engine)
So now you have two options:
putting smaller jets in your carburetor until your air box supplies the right amount of air. (check your spark plug color) As a result, your moped will make more miles per gallon of fuel, perhaps at the cost of a few mph. (do you have a variator? I don't and my top speed is governed by the rear pulley more than by jet size/air supply)
increasing the amount of air that the air box can supply.
if you do this, I'd start with drilling more holes behind the metal gauze filter. There now seems to be only two holes there, my original stock filter had three holes there. These newly drilled holes will not increase the noise your moped makes. (i drilled those holes in a hexagonal pattern)
If there's no improvement in speed, you can drill an extra set of holes in the outer shell but this will increase the noise if you don't try to mimic the current setup. so if you drill a hole, you should glue a pipe onto it. (I made them from a few inches of a rubber fuel tube) Then the engine sounds stock and you have more air intake.
My air box looked a little like the attachment and it was very easy to drill three more holes and put on three pipes on the other side. (mirror image)
I can't see if your pipes run through the center or not so in your case it might be more difficult.