Patch Including Application Actions

Action is an Application configuration not an object on its own. Meaning you can't patch an action in an Application.
You can still patch the process model and add a post deployment step to configure it as an action in the target environment.

If you don't want a post deployment step you can choose one of the below :-

1. Export and Import the whole application - be sure of all changes that would go in with this deployment.

Or If it is not too late

2. Create two packages - one for the action(comprising only the process models only) the other that consists of all other helper objects (All objects).
This way going ahead if you need any configuration changes, you only deploy the published application that has actions, security, Application Name and Description.

I don't think either of these options will work. Instead, since we are using AVM to push our patches to our git repo, I just made a change to the /application/<applicationGUID>.xml file. This was a rather messy and likely unsupported way of doing this.