My nose is running. I am hard to breath through nose.
My nose is running. I have a difficulty breathing through nose.
Which is the correct adjective to describe that you have a runny nose.
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Both could be OK with some editing:
My nose is running. It is hard to breathe through my nose.
My nose is running. I have/am having difficulty breathing through my nose.
A possessive pronoun is usually used when describing someone's body parts.
As a note: Some people might use a difficulty, though I wouldn't.
In everyday conversation, I think I'd say it something like this:
My nose is running; it's hard for me to breathe through my nose.
My nose is running; I can't breathe through my nose.
(In the latter case, "I can't breathe" is being used to mean, "I'm having trouble with breathing" – it doesn't necessarily have the more literal meaning of being unable to breathe through my nose at all. English is tricky like that.)
Difficulty is a more formal word that; normally, I'd be less likely to use it except in a more formal context. For example, a doctor might ask a patient:
Are you having any difficulty breathing through your nose?