Breaking

Does the clunking noise only happen once?
If not, does it slow down with wheel speed?
Does it hesitate while coming to a stop or when idling as well?
If it does hesitate when you come to a stop, how long before it idles smoothly again?
Does it hesitate when you take off from a stop?

I will watch for your reply and get back with you ASAP?

The clunking noise could be something in the front suspension such as a ball joint or tie rod getting ready to fail. Do you notice any handling problems, such as wandering pulling to one side when braking or uneven tire wear?
I am not sure if the Cutlass had rear drum brakes that year.I don't think so. However, if it does there could be in drum hardware components that are rattling around.
Or the front brakes could have something wrong, maybe even a loose mounting bolt.
I know this does not jive with the hesitation, but when brake or suspension are in bad enough shape, it can feel like that.
The biggest worry I have is safety, that is why I wanted to bring them up first.

Something that is more feasible is the power brake booster. If it has failed the diaphragm can make noise and the loss of vacuum will make the motor hesitate.

Here is a quick test to see if the booster is bad;

Before starting the car pump the brakes until they are hard and hold the pedal.
Then start the car.
The pedal should fall down some under the pressure of your foot.

This is not a definitive test, but it it is a quick way to check.

More than likely it is the brake booster or a leak in the vacuum hose from the manifold to the booster itself.

Let me know how it goes.