I absolutely sympathise OP, I face similar types of situations on a regular basis (single parent, 3 teens, good job but always trying to just about manage and give teens as much support / opportunities as I can). So I'm not saying what follows out of any sense of superiority.
Your wider situation is unsustainable - your business model isn't working if a delayed payment means a stark choice between eating and a music lesson. Regardless, you need to start to tackle this.
You also need a back-up like an overdraft or credit card - you don't have to use either regularly but you should have them for these or similar eventualities.
I don't agree with posters suggesting you should stop the lessons entirely - extra-curricular activities are important, not of course more so than food but should be maintained if you can, and you need to plan better.
In terms of this specific situation, I definitely wouldn't feel able to tell the teacher (I can see why people say you should). I would use any money at all that I have, even savings or money for another payment, resolve when you are paid, and then commit to better forward planning in future. For example, when you sign your DC up to an activity, you know how much it will cost and should budget ahead for that, so that you don't run into the pinch points.
As I said, I say this as advice I have to give and try to take myself - what happens to me far too often is that while I have budgeted for known expenses with a little wriggle room, I often have no way of managing an unexpected situation e.g. like a car repair or school payment I didn't forsee. This isn't a good way to be and I have to keep trying to work on it.
I think you should do all in your power to keep the lesson and pay whatever way you can.