This is a case for the LC29H.

You can use it with direct USB to your android phone using the app “SW maps”. Make sure you configure the jumpers to USB mode (A and B together) so you can connected it to your phone. There's no need to use a Raspberry PI (which is hard to configure and the documentation is not that well created).
You'll also need a NTRIP client and connection to the internet while reading your location. There are a lot of free and trial NTRIP clients that you connect to. Probably your country state offers free access to theirs. Quick search “[your country here] free RTK and NTRIP correction services” on Google will solve this :)
If you live in Europe, take a look here: https://ntrip-list.com/europe/

This is the module in use:
LC29H RTK high-precision positioning GNSS (GPS) module
https://s.click.aliexpress.com/e/_c4mB6hpl

USD adapter (micro USB to USB-C)
https://s.click.aliexpress.com/e/_c31bGSZt:

4units M3 10mm Flat Head screw:
https://s.click.aliexpress.com/e/_c2uSrZ3t

This LC29H module has 3 versions:
AA - This is a generic GPS with bit more precision (but not much). Used to track locations with your antenna with 1 meter error.
DA - Used to track locations (using RTK) with your antenna but with 1 centimeter error.
BS - Used to create a Base Station. The Base Station is used to increase accuracy of your positions. You basically create your own NTRIP server with it. This module can't do anything by its own. If you have a Base Station you will need a DA module to operate with each-other. I haven't explored this BS module, but as I found out, you'll need to connect it to a computer (or an RPI) in order for it to calculate the corrections that are then sent out to your tracking logs.

Note that a DA module can operate by its own without BS module. But if you have a BS module, you can increase your accuracy. If the field is a closed environment or the heather doesn't help, your precision will reduce drastically. If you have a Base Station, you can keep the 1cm precision without good conditions.