To pule is to cry, but not loudly. Puling is weak, soft crying. It's a pathetic sound.
Unfortunately, there are many ways to cry. You can bawl, which is loud crying. On the other end of the spectrum, you can pule. When you pule, you're crying weakly at a low volume. When dogs are scared, they often pule. Children sometimes pule when they don't get their way. People pule when they're sad, frightened, heartbroken, or tired. People pule when they don't have the energy to cry louder.