Perform the following steps:
git submodule init
git submodule update
cd gdal-src/source
git pull
git checkout v3.8.3 # corresponds to the tag you want to update to
cd ../../
git add gdal-src/source
git commit -m "Update bundled gdal version to 3.8.4"
These steps assume that there are no fundamental changes to the gdal build system.