If you are at a drop in visit longer than a half an hour will the client get charged more?

No, the client will not get charged more. You can stay longer if you'd like, and it won't affect the 30 minute agreement. I overstay my stay too with my very very regular clients, and instead of 30 mins I give them 45 or 60 minutes when I feel like it.

You will only be paid for the agreed upon booking rate, if your visit requires more than 30 minutes you should discuss this with your client and make adjustments as necessary