snuffle
To breathe noisily through your nose, often when it’s stuffy.
To snuffle means to breathe noisily through your nose, especially when it's partly blocked by a cold or when you've been crying. When you have a stuffy nose, you might snuffle instead of breathing clearly. You can hear someone snuffling when they're trying not to cry, taking quick, shaky breaths through their nose.
Animals snuffle too, but for different reasons. Dogs and pigs snuffle when they're exploring with their noses, making soft snorting sounds as they sniff around for interesting smells. A dog might snuffle along the ground searching for a lost toy, or a pig might snuffle through leaves looking for food. Their snuffling is perfectly normal: it's just how they use their powerful sense of smell to investigate the world.
The word captures both the sound and the action. It's somewhere between a sniff and a sniffle, with a bit of snort mixed in. When you're getting over a cold and your nose is starting to clear up, you might spend a day or two snuffling before you can breathe normally again.