We’ve all had cheap whistles with loose mouthpieces.
The standard remedy is to use Teflon plumbers tape to snug things up, but there are times when it is very difficult to use this method.
When only a very slight gap needs to be filled between the tube and the mouthpiece, plumbers tape just doesn’t want to stay in place when you try to slide the mouthpiece over it.
Try striking the inside of the top end of the tube with a nail set… but be careful to only strike hard enough to raise a small bump on the outside of the tube, and don’t punch all the way through!
Smooth the bump with steel wool if necessary, and there you have it.
Now the mouthpiece should fit snugly on the tube without the need for plumbers tape.
I have used this method to snug up a loose tuning slide on a Dixon with the brass tuning slide. You MUST be sure to use a fairly solid backing when doing this so as to not make the tubing out of round.
For the tuning slide I sanded a 1/2 inch dowel until it just fit the hole and then tapped from the outside in three places. Start light and work your way to deeper dimples. This left just enough friction.
Another way is push and twist a closed pair of needle nose pliers into the open end of the tuning slide, slightly flaring it.