HI Abbey,
yes, you can re install MacOS on your Mac mini.
I would recommend Upgrading the internal Hard Drive (HDD) to a significantly faster Solid-State Drive (SSD), such as this one: [link|https://www.westerndigital.com/products/internal-drives/wd-green-sata-2-5-ssd?sku=WDS240G3G0A|WD Green™ Internal PC SATA SSD Solid State Drive SATA III 2.5"/7mm Cased|new_window=true]
The easiest way of fully wiping your existing hard drive will be to reformat it. You could possibly do this within disk utility however you can also use a different device. Just make sure to format it as MacOS extended journaled and back up any important data you wish to keep. if you purchase a new drive, you will still have to format it to MacOS extended journaled.
This system can run the last version of Mac OS X 10.6 "Snow Leopard". However, it cannot boot in 64-bit mode, nor does it support "OpenCL". It also cannot run versions of Mac OS X more recent than 10.6.8.
Here is a link to download the ISO file: [link|https://archive.org/details/snow-leopard-install_202112|Mac OS X Snow Leopard 10.6 (10A432) (Retail DVD) : Apple Inc. : Free Download, Borrow, and Streaming : Internet Archive|new_window=true]. Please note you may have to try a few different sources of Snow leopard 10.6 before you find one that will work.
From here, you will need to install a piece of software such as Balena etcher: [link|https://etcher.balena.io/|balenaEtcher - Flash OS images to SD cards & USB drives|new_window=true].
This will allow you to flash an OS .iso File to a USB stick which is what you will use to re download Mac OS. Here is a guide on how to do this: [link|https://www.youtube.com/watch?v=1p-r9rqxpic|Balena Etcher how to make a bootable USB. (MAC OS) (youtube.com)|new_window=true] You will need a USB stick with a capacity of at least 8GB.
Then just make sure your Mac mini is fully powered off, insert the USB drive, Press the power button and quickly press and hold command+R. After this, you should be able to simply go through the installation steps to re-install MacOS.
I hope this helps, let me know if you have any other questions :)