Assuming the school has the ability to add it to the budget…
Baseball mound clay is sold in bags. It is the absolute best stuff to tamp into the low spots on the batters box and in the pitchers circle. I see so many places take the loose dirt and try to tamp it in and it lasts about one half inning before it’s a huge hole again.
Broom all the loose material out of the hole. Lightly wet the ground that is now free of all loose dirt. Then pour the mound clay into the hole and tamp away. It will last entire games that way. Eventually the batters box and circle will be all clay and will hold up incredibly well.
If the holes are really big you can do this same thing with clay bricks.
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Yes, we use that. I was amazed how well it works. I was very skeptical at first, but applied it per the instructions, and it was like a new field.
Being in GA, our fields are clay. So whatever I drag is mostly clay and very compactible and similar to that product above. If it was more of a dirt field, I wouldn't waste my time tamping it down.