Short answer
Yes, plain cooked crawfish tail meat is safe for cats. The meat itself is not toxic, and a small amount in a broth is fine. Just be sure the pieces are free of seasoning, butter, or garlic. Offer fresh water, monitor for any vomiting or diarrhea, and call your vet if anything seems off.
Crawfish — Food safety facts
Safe to share · Cats · full breakdown on the linked food page
Plain cooked tail meat is safe. Cajun or garlic-butter versions are not.
The full verdict, portion guidance and serving notes are collected on the Crawfish food facts page.