Is this correct? I thought the national government provides the local government the funds based upon the number of preschools in the area.
No, they are entirely dependent on local and occasionally provincial governments. The funding situation depends very much on whether they've got the finances, as there were plenty of documented cases this year where privately run kindergartens no longer received the educational subsidies.
Have you spoken to the person responsible for education in the Gmina to ensure that you'll get the funding? This is Poland - it may well be that you'll be expected to give the director's job to a certain person in exchange for the funding, especially if you're already aware of problems with corruption there. It won't be official, but you can expect strong hints that they would prefer to see a certain person in charge.
I'd also verify that the funding is already in place for other private kindergartens - if you will be the first one, then they may simply refuse to subsidise it in any way.
If you want, I'm happy to have a chat with you about this in detail.
To give you an idea - DGP (one of the most reliable newspapers in Poland) published this article last year - serwisy.gazetaprawna.pl/samorzad/artykuly/1451502,przedszkola-walcza-o-dotacje-w-sadach.html - there's a total of 17 *billion* PLN in claims against local governments from the operators of private kindergartens.