Yes!
It it's completely fine to hear your breaths when singing! Hearing breathing in a song is perfectly fine. Sometimes singers will hear a recording of themselves singing, and hear their breaths between phrases, and think that that is wrong or sounds bad.
But actually, hearing a singer's breaths in a song is what makes it sound natural and real. If you take those breaths out, it would make the vocals sound very artificial, like a robot was singing, or someone chopped up the vocals.
If you're thinking to yourself, "but I never hear singers' breaths when I listen to their songs."
Yes, you have!
You just weren't focusing on it.
To prove this, check out this Adele song, "Set Fire to the Rain".
Listen to this specifically for Adele's breaths between each phrase she sings. Notice just how much she breathes! When you are paying attention to it, you'll hear how loud the breathing sounds actually are.
Test it out -- pull up your playlist, and choose a song at random. Listen to the song specifically for the breaths. You'll hear them instantly. Some singers breathe a bit heavier, some breathe a bit quicker, but you'll hear them!
If you were to take those breaths out, again, it would end up sounding very cold and unnatural.
So, please don't be worried if you're hearing breaths on your singing recordings. You're supposed to!
As always, I hope this post helped you! Comment with any questions or with your successes, and if you need any extra help, you can grab an online lesson with me over here!
Happy Singing!