Clean URLs for applications

Probably what you want to do is enable the core module called 'Path'. This will create a new menu item under administration->site building called 'url aliases' which allows you to set custom path names that alias the underlying /node/number format. HTH.

I already have that enabled, and all the paths to other node types are working fine. In fact the single job paga also has the correct alias. But when the user clicks on "submit an application" - that's the page I am talking of.

Works for me. Did you create an alias for the application page itself? For example, on my test system I did this:
job/application/node/1 -> /apply/test-job-1
After clearing cache and reloading the job_posting node page for 'test-job-1' I get the desired url alias. Kind of tedious for every node but it definitely works here.