The plastic cap has a recess that the metal duct goes into. The duct usually has protrusions all around the circumference at the end that keeps the duct firmly in the cover. The duct needs to be circular not an oval to insert it. It might be easier to buy the cover with the duct and replace it as a unit.
Ron

Thanks Ron for the response.

I do not have access to the inside ductwork to replace the cover with one of those cover duct unit jobs your mentioning. The duct run at the back end of a crawl space.

What you need to do is purchase the hand tool that makes crimps in duct pipe. Crimp the end that you can get to then the dryer cap replacement will slip on.

Here is a link that might be useful: pipe crimper

If I am able to get in the back of the crawl to fix it, would my idea be an acceptable fix.

Duct runs about 30-40 feet to out the wall termination. I would cut existing ductwork at the termination. Feed new vent cover unit in. Connect existing rigid duct with new vent cover unit with duct tape.

Is using duct tape a bad idea at the very end of this exhaust run?

Use HVAC foil tape instead of duct tape. Duct tape is used on everything but ductwork. The foil tape is available at the big box stores in the tape section.