Is it common after a partial knee replacement to experienc a clicking sound when you bend your leg?

No but its okay. Painless clicks in the knee are not a concern. Painful clicks are investigated. Clicking early after partial knee replacement can occur in some patients but usually goes away over a few weeks or months. It is usually occurring from the metal femoral component tapping against the polyethylene plastic bearing when you go from the swing phase of gait to heel strike. Heard mostly by the patient.
Possible. Clicking after replacement is generally a function of the patella either jumping off the femoral component or impingeing agaist it but usually decreases after some time.