The title says it all, and I'm overdue to address this question - it's SO important to my teaching philosophy, and I'm happy to discuss in-depth.
My goal as a teacher is to impart a sense of excitement and understanding of music to all my students. My goal as a teacher is NOT - and never has been - perfection. As my favorite fitness instructor says, "there's no such thing as perfection." When it comes to music - especially with younger students - an emphasis on perfection can lead to lost interest, frustration, and unwillingness to "play" music.
I focus on PROGRESS, not perfection. What this means is seeing a student make progress in some way each time I see them - whether I can tell they've practiced more, they're more excited about a song, or they really grasped a particular concept - is much more important to me than whether they can play a piece "perfectly."
There is a difference between mistakes and errors when it comes to learning music. For example, although I'm a professional musician and longtime music educator, I still make mistakes all the time, especially when I first sit down to practice each day. Mistakes happen when we're rushing through a piece we haven't properly prepared for, or we aren't focused on what we're doing, or we just happen to hit the wrong note.
Errors, on the other hand, are consistent. In my studio, I do correct errors. Errors are when a student consistently misses a particular note, or doesn't understand/hear a rhythm, or struggles to identify specific markings/notation.
How do I tell the difference? Usually my first suggestion to a student who's made a mistake or two is to try the piece again, more slowly. If all the mistakes disappear, we're good go! If not - and it's a very specific, consistent error in the same place, every time - then we address it and spend some more time there.
So what happens if we've addressed something in-depth and the student is still struggling? It depends on the student. Some students are absolutely fine spending extra time on a concept to really get it down, and some students lose interest after a few passes. There's more than one way to learn music, so if we're in the latter situation, I'll usually move on, and reintroduce the concept with another piece later on. There's no benefit to hammering an idea after a student has mentally checked out - and it takes the enjoyment of playing away from them.
In short, as long as your student is practicing consistently at home - playing correctly 3-5x a day, most days - we're in good shape. We're looking for progress, not perfection, and those 15 times per week playing the song will definitely create progress. Celebrate how far your student has come, and remember to praise effort over perfection to help keep students engaged at home.