Thanks for using Just Answer. I will be glad to assist you.
I am not sure if I understand your question.
Does the starter engage for a couple of seconds and then stop turning the motor?
Do you hear a single or multiple clicks and where do they come from?
I also believe that the noise is coming from the solenoid.
The system does not have a starter relay, which is a common issue on other vehicles with these symptoms.
Please refer to the diagram below:
There is a Technical Service Bulletin that may be related to the issue you are having. Please refer to the information below:
Bulletin No.: 83-64-16
Date: June, 1998
Subject:Weak or Discharged Battery, Slow Starting, GEN or MIL Lamp On, Various Driveability Symptoms (Repair B+ Stud and/or Wiring on Starter)
Models:1997-98 Buick Skylark1997-98 Chevrolet Cavalier, Malibu1997-98 Oldsmobile Achieva, Cutlass1997-98 Pontiac Grand Am, Sunfirewith 2.2L, 2.4L, 3.1L Engine (VINs 4, T, M - RPOs LN2, LD9, L82)
Some owners may report the battery is weak or discharged; the starting motor may turn slowly; the generator is not charging and/or the generator Malfunction Indicator Lamp (MIL) may be on, possibly intermittent. Driveability symptoms may also be experienced.
If following steps 1, 2 or 3 of strategy based diagnostics did not reveal the cause, the generator output and battery positive cable terminal at the starting motor (CKT 1 - VIN T and M; CKT 2 - VIN 4) may be loose, causing a high resistance in the battery positive circuit from the generator to the battery.
Inspect the condition of the starting motor B+ stud, wire, ring terminals and nut. Clean each mating surface and replace damaged parts. Tighten positive battery B+ cable to starter solenoid nut to 12 N.m (106 lb in.). Refer to the "Starting Motor Replacement" portion in the Engine Electrical Section of the Service Manual.
Recharge and load test the battery if required. Refer to "Battery Charging" in the Engine Electrical Section of the Service Manual.
Important: The generator and battery are not damaged and should not require replacement.
If all of these test check fine, we will need to get the starter out so that you can have it bench tested.
It is easiest to take the starter to an autoparts store and let them bench test it for free. But if you would like to test it then here is the procedure:
IMPORTANT: The current will decrease as the windings heat up.
Can you find where the clicking noise is coming from?
Maybe the relay box or under the hood. Please isolate the clicking noise and let me know where it is coming from.
Also if you could check for spark that would also be helpful. Have someone start the engine and leave the spark tester on as it dies. See if it is dying because of a loss of spark.