I will try to find some hours elsewhere, but most schools want to pay me under the table and I need legal, above-board hours.
Why not simply give a few private lessons and give them receipts? You could even give them 2x20zl receipts rather than 1x40zl to bump the hours up a bit, as well as getting the school to pay you (on paper) for twice the amount of hours worked for half the money.
Having said this, I'd tell the immigration people without hesitation. I've got absolutely no time for illegal immigrants - white, black or yellow, they're still illegal.