Reinstalling OS on Mac mini

We need exact model information. The alphabet soup model information you provide is somewhat lacking in specifics and detail. :-) Assuming this isn't a PPC G-series Mini it is quite possible you need to buy a Snow Leopard installer disc.

Snow Leopard disc source - https://www.apple.com/shop/product/MC573Z/A/mac-os-x-106-snow-leopard

Snow Leopard 10.6.8 Combo Updater - http://support.apple.com/kb/DL1399?viewlocale=en_US&locale=en_US

It is advisable to use Software Update at least once so you get special updates such as the "Mac App Store Update for OS X Snow Leopard" - https://support.apple.com/HT205702

For questions in English please post in the English language community. You will get a more speedy response, in English, from a larger community that's active 24/7. --> https://discussions.apple.com

If it is from 2007 then likely it is an early generation Intel Mac and can run OSX up to 10.7.5. I am pretty sure Apple still sells Snow Leopard DVDs (of course it up up to the OP if purchase of such is desired). You do not have to have a DVD with "this OS [OSX 10.4.10]". In fact many still regard Snow Leopard (10.6) as one of the pinnacles of Apple system development and I know I have had several people ask me recently if I would sell them my unopened Snow Leopard server DVD.

Some of these older Minis are still desirable as media servers. They can hold multiple rotational drives with large storage and you don't need lightning speed for that.

I think the Mac mini ist from 2007 and you don't get a macOS for such an old mac. The Mac Mini was delivered with Mac OS X 10.4.10. If you doesn't have a CD or DVD with this OS you can't do anything.

Here you can finde the answer:

English: https://support.apple.com/en-us/HT204904

Deutsch: https://support.apple.com/de-de/HT204904