Hello hazelpa,
We have a great support article that might help with this. Please take a look at it here: If you can’t send or receive messages on your Mac - Apple Support. Follow any steps you've not yet taken from this guidance:
"Try these steps first
After trying each step below, try to send a message again.

If you see ‘This user is not registered with iMessage’
If you try to send a message and see an alert that the person isn’t registered with iMessage, the message is trying to send using SMS/MMS. Make sure Text Message Forwarding is set up and your iPhone is powered on.
If your contacts see your email instead of your phone number
If you can send messages but your contacts see your email address instead of your phone number, make sure that you're signed in to iMessage with the same Apple ID on all your devices. Learn how to add your phone number to Messages on your Mac.

Learn more
We hope this helps. Let us know if we can assist with anything else at all. Take care.