When you look at something blearily, your eyes are not doing their job. You have just woken up, you are very tired, or you have been crying, so the world looks soft, watery, and not sharp. People most often put it with look, stare, or peer: you blearily look at the clock, or blearily try to read a message on your phone. It is not a doctor's word and not slang; it is an ordinary, slightly story-like way to paint that half-awake feeling. Though actually, speakers stretch it a little: if you are only blearily aware of something, your mind is as unclear as your eyes — you half-know it, nothing more.

睡眼惺忪地、迷糊地（因疲倦、剛醒或剛哭過而看不清）