The phrase "begging dog" refers to a dog that is asking for something, usually food, by giving puppy eyes, wagging its tail, or sitting close to someone. Full definition
I also don't appreciate one of their forum mods coming online and talking down to the fans like begging dogs or mindless children. (thedreadpirateguy.com)
:) Here are myths about begging dogs dispelled plus tips on preventing, fixing and using the behavior to your advantage. (animalbehaviorcollege.com)
Another important step is to ignore a begging dog while you eat. (veterinarianedu.org)