Here's why:
* TPS Failure: A malfunctioning TPS can send erratic signals to the engine control module (ECM). The ECM, trying to correct what it perceives as fluctuating throttle input, might be causing the servo motor (or other mechanism controlling the throttle cable in that era of Cadillacs) to repeatedly adjust the throttle position, resulting in the clicking and movement you describe. This is particularly likely if the clicking is accompanied by rough idling, stalling, or poor acceleration.
* Wiring Issues: A short, intermittent connection, or damage in the wiring harness to the TPS could also mimic a failing TPS. The intermittent nature of the problem (20-30 seconds) suggests a connection issue is a strong possibility.
* Less Likely (but possible): While less probable given the duration and repetitive nature of the clicking, a problem with the throttle cable itself (kinking, binding, or a damaged cable housing) *could* theoretically cause this, but it's less likely to produce the rhythmic clicking.
What to do:
1. Safety First: Don't drive the car extensively with this problem. Erratic throttle control can be dangerous.
2. Visual Inspection: Carefully examine the throttle cable and its connections. Look for any signs of damage, kinks, or binding. Check the TPS for any obvious physical damage.
3. Professional Diagnosis: The most reliable way to diagnose this is to have a qualified mechanic, preferably one experienced with older Cadillacs, perform a diagnostic scan using an OBD-II scanner. This will check for any error codes related to the TPS or other throttle-related systems. The scan may indicate the TPS is sending erratic signals or there are other problems detected.
4. TPS Replacement (if necessary): If the diagnostic scan points to the TPS, replacement is usually the solution. The part is relatively inexpensive and the replacement procedure (though it varies slightly depending on the year and model) is generally straightforward for a mechanic.
Ignoring this issue could lead to further damage or a more serious problem down the line. Have it checked out promptly.