For info these commands helped me reset and retry getting the subtree stuff to work
# removing the subtree from existence
git filter-branch --index-filter 'git rm --cached --ignore-unmatch -rf public' --prune-empty -f HEAD
# removing the gh-pages from the remote
git push origin --delete gh-pages
When you setup a new gh-pages branch following the hugo documentation everything is as it should be and works. You create the deploy script and it works. Then you go into github and set a new url for the github pages site. Github does this by adding a file called CNAME into the gh-pages branch root. This commit then prevents a subtree from working in all the attempts I have made.
Every time I came to do a git subtree push it failed stating there were changes on the remote which were not in the local and it would be correct. Doing a git subtree pull did not fix it though and so the only solution I could think of doing was handling the creation of the CNAME file manually. Now the deploy script works without a hitch!