Nosebleed
Bloody Nose
noun
A nose with blood is a nose with a nosebleed. It is a bloody nose.
The readings are the kun'yomi vocab readings. You already know 鼻's reading from the kanji and vocab, and you should know the vocab reading of 血 as well. Put them together and you have はなち, though you want to change the ち to ぢ. Alternatively, it's possible (though less common, but it doesn't matter if you're turning it to kanji via an IME anyways) to write it as はなじ. I guess that blood added a little tenten (it represents the blood coming out of that nose!). Spurt spurt spurt.
私は鼻血がよく出ます。
I often get nosebleeds.
急に鼻血が出てきて、びっくりしました。
I was surprised when I suddenly had a nosebleed.
鼻血が出ちゃったんですが、ティッシュってありますか？
I have a nosebleed, do you have any Kleenex?