Hello @Leo Keogh - The doc states the following on publishing to a private registry:
And the Pull should occur automatically after the detection of a newer image on the registry. Is the issue you're facing the same authentication not working when pulling? If so, please try running the following after the creation of the private image:
az webapp config container set -n <functioname> -g <resourcegroup> -c <yourCONTAINER_PATH> -r <registry_URL> -u <username> -p <password>