Probably the most tested and true trick to respond to the increased requirements is to create a "Phase II" in the project. You can easily respond to your client, "OK, we will process this, but in Phase II, let's focus on the current requirements to be able to finish on time".
In some cases, clients will object to a phase II, and in this case, you have to learn how to say no. If you keep on saying yes to every incoming change request then your project will most likely fail.
In theory, what is happening is scope creep. However, I have my own specific definition of scope creep, which means that scope creep consists of new requirements that should've been in the project in the first place, and were not that mainly due to poor requirements management (or, if you like, bad project management).