Q.ponytail palm offshoot
Can I take a cutting, not a ‘pup,’ from a Ponytail Palm tree? I have a Ponytail Palm that is about 8 or 9 foot tall. It has an offshoot growing out of the trunk stem that is about 2-3 ft. tall. Can I cut the stem and transplant it?

I would not advise any attempt at removing this off shoot of your Pony Tail Palm.
I believe this to be a pup that was allowed to continue growing. Your palm looks healthy and I would allow it to continue growing as it is.
I have listed a link about Pony Tail Palm pups.
https://www.gardeningknowhow.com/houseplants/ponytail-palm/propagating-ponytail-palm-pups.htm