Delicacies increase obtained dog upbringing base experience. They can be bought in the shop or found during a walk. There are the following delicacies: sausage, cheese, Cookie, bone.
More detailed information on delicacies can be found here

Bonuses from the delicacies found during a walk are being summarized and added to the bonuses from the bought delicacies. If the delicacy, which was found during a walk, has a higher bonus level than yours, then your delicacy bonus will be increased.