On a large speaker it is "just a dust cover" but still needs to be there to separate the front of the speaker from the back. Without it there would be much less sound.

But in this case it is a tweeter... It is more important on tweeters and is not "just a dust cover", it is the actual cone that causes the sound and is made that shape for a reason.

Masking tape works really well, you need to smooth it out so it gets maximum contact in the centre of the hollow and pull it back out.
Often times poking things in the back of a speaker won't work as many of them use a mesh cover to stop just a thing from happening.....

what he said ....seems like a good idea....a straw or something