Is there a term for letting out an exasperated sigh through the nose?

"Huff" would be the word that I have seen used for this. It was difficult to come up with a list of citations - The surnames Huff and Huffington cluttered up the results. A. A. Milne apparently believes that snails "huffle" when they are in danger. That's a small joke - see his poem "Four Friends".

Perhaps humph

\a snort articulated as a syllabic m or n with a voiceless onset and ending in a nasal h or a glottal stop; often read as ˈhəm(p)f\ Definition of HUMPH

—used to express doubt or contempt

This nasal interjection is distinct from the throaty interjection, harrumph

• to clear the throat in a pompous way

• to comment disapprovingly

which can be verbal or on-verbal.

both Merriam-Webster

Although dictionaries do not explicitly support it, writers of closed captioning for film and TV seem consistently to have settled on the verb scoff to indicate this specific manner of expressing contempt or irritation. (The word more generally signifies expressing derision by whatever means.) Flaring one’s nostrils is an associated physical sign, though of course that can be done without any such sharp exhalation as is here inquired about.

It depends on the type of sound that comes with the air blowing out of the nose. If you're looking for a word that describes the same kind of soft sound that comes out of the mouth when a person sighs, I would call it the same word--a sigh. A snort makes a completely different sound and is also used to describe inhalation. I can't think of another word that makes the same kind of soft sound, so I would apply "sigh" when it comes out of the either the mouth or the nose.