The phrase "soft mouth" means being gentle and delicate when holding something in your mouth, often used to describe well-trained dogs that don't bite down too hard on things. Full definition
She enjoys getting treats and has a very soft mouth. (healthypawsla.com)
The shaping of soft mouth is done with the help of the treat. (snowyswan.com)
They have been given an especially soft mouth that helps them to even hold a raw egg without breaking it, perfect for retrieving the fragile fishing nets from the water. (dogspot.in)