Hi. Imagine having developed a classical website with java&struts. Now you customer is learning that redeploying the application to change an image or a text is a significant cost. And it asks to add a function to the site: cms-like handling of the contents (editing, versioning, approved publishing).
How would you handle this request? Would you develop it in the webapp? Would you merge the webapp with a CMS? Would tou MOVE the webapp into a cms? Would you run away?
Building in simple CMS support isn't too difficult (provided putting the authentication and security isn't too stressful in your framework) but it can get really fiddly if you need to provide a whole WYSIWYG environment where they can upload content and format it.
If they want the whole lot, I'd consider rebuilding inside an existing CMS. If they can cope with simple edits, build that into your app.
But the thing that'll sway it for them is the cost. You need to let them know they're paying for you to re-build (and give them a quote so they know how much it's going to cost them). You can't swallow that cost unless you massively misunderstood the brief.
If they're not happy with what you've got so far but don't want to pay you for building it up further, explain you're happy to bring the original brief to fruition but invoice them for your time so far (or percentage of the contract if it's project based) and let them think about it.
If you cocked up the brief, you either walk away not or partially -paid and lose a customer, or take it on the chin and try to do better next time.