I would just like to point out that those corks are there for three reasons. The first is to hold the mute inside your bell, the second is to protect the finish on the inner bell from being scratched by the metal of the mute, or vice-versa. The third and perhaps most important is to allow the air a way of getting out of your horn.
If you look at the other end of your mute, you may notice there is no hole in it. So if you remove the corks, then put the mute in the bell, there will be no way for the air to escape the bell. This would either stop you from playing the horn at all, or you might blast the mute out of the trumpet like an air cannon. ;)
The practice mutes that were mentioned do indeed seal completely around the inside of the bell, but they also have a small hole in the other end to allow for airflow. They are a bit restrictive to play, but they do accomplish the job.
The mute in your picture is a performance mute, and while it makes the sound of your horn a bit "tinny," as you have mentioned it does not really quiet it much.
I think in your case what you want is an actual practice mute; Humes & Berg's Stonelined Practice Mute for trumpet is fairly effective and pretty inexpensive.
I would suggest seeing if the store you bought it from will allow you to exchange it.