How do I install local application dependencies in a Kubernetes manifest?
You can't add run-time dependencies on an already build image using a yaml manifest. 
You might attach/exec to an already running pod and install them, but it makes no sence, since you need to make sure that when the pod is re-created you do it again. The much better and right way is simply to describe it in your Dockerfile. 

Kubernetes is not created for such kind of hacky moves. It simpy maintains state that you describe.  Essentially you need to add an instruction that adds git in your Jenkins dockerfile, build your new image and refer to it inside the manifest.

Commands and args are used if you want override the image's default entrypoint and its arguments.