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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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.
or even:
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?
For me, difficulty breathing and hard to breathe are too general to capture the just the sense of My nose is running. Instead, except perhaps in the most formal contexts, I would simply use runny.
My nose is running. I have a runny nose.