If you push down on one side of the cone, you'll askew the cone in a very unnatural way.
You ESPECIALLY shouldn't push anywhere on your cones while they are in operation, because you could "rock" the cone assembly causing it to strike or rub the side of the pole piece, or the side of the gap.
REALLY bad idea.
Of course, if you push down on one side of the cone, you'll tip it that way, and you'll potentially cause the former to rub on something that it shouldn't.
That's not a bad sign...
That's an "of course".
If the Brahma is an older version, you might be experiencing lead slap - not against the cone, but against the outer roll of the spider.
The cheap and effective homemade solution for this is to just glue a small layer of felt (similar to what's glued to your cone) to the outer roll of the spider, just below the leads. The one I have here has a cotton ball stretched and glued in place, and it worked.
You might also look to where the tinsel leads are soldered to the metal tabs that go to the speaker terminals... sometimes you can simply gently bend these back a bit - or maybe in your case they are bent too far...
That is the component that Adire had tweaked several times in an effort to fix the lead issues, the angle of those, and the actual length of the tinsels.
One other possibility...
If you've really beaten on this thing (if it's even in a medium enclosure, even 800 watts could do this), you could have overexcurted the driver... if you pulled the leads taught when you did this, you may have pulled them out from under the dustcap a bit, essentially leaving them too long under the cone, which would definitely cause lead slap. In that case, some servicing is in order.