This in fact is not a spoken word so much as a sound. It is the same sound made when someone is sniffing their nose as they are crying or trying to stop crying. The recovery sniffs is the sound you are mocking.
It is a sarcastic way to show someone you feel no pity for their situation.
Just like playing the tiny violin for a person, this mocks the persons sadness by "crying" along with them. See example
Bjorn: Yesterday, my girlfriend told me that my clothing is out of style and it really hurt my feelings.
Bennett: (very sarcastic)"Sniff-Sniff".... "Sniff- Sniff" oooh "sniff-sniff"...that's just terrible
Bjorn: Oh shut up