Citation 680 flight crew reported while on base leg the aircraft experienced an engine malfunction which resulted with the flight crew using idle power in one engine and landing safely at the airport.

Narrative

On Day 0 we entered the uncontrolled pattern at ZZZ. On base leg I requested flaps 35. The aircraft started shaking and vibrating. There was an audible banging and chugging. I glanced inside and saw the left Interstage Turbine Temperature (ITT) tape had turned amber and ITT was rising rapidly. I immediately reduced the left thrust lever to idle and the temperature came back under limits. The highest we saw was 932. About this time we were turning final maybe 3 miles from the runway. After a quick discussion with my partner; we decided the safest course of action was to continue and land. I did say I wouldn't use the left thrust reverser. We landed uneventfully and taxied in. After taking care of the passengers; calling maintenance; the Chief Pilot; and Operations Supervisor; scheduling put us on duty at the FBO for 2 hours and then gave us a hotel. I called at the end of the duty period for shutdown and they told me they were bringing another plane in for us in a few hours. I believe it would have been 3 to 4 hours after our landing.In our minds; we essentially had some sort of engine failure. Mentally we were already in the decompression mode once they gave us the hotel. I called scheduling and said we weren't flying anymore today and we were done for the day. I said it was because of the engine problem we had. Scheduling; or at least the person I talked to seem to be having a hard time understanding what had transpired. After some discussion he asked what time I wanted to show at the next day. I said XA:00 local. I got a request to submit a report. I never used the work fatigued. We were not mentally prepared enough to go jump in another airplane.As of Day 13; the plane is still in maintenance at ZZZ.When there is a serious malfunction; the Chief Pilot and or scheduling should immediately remove the crew from flying that day. The company needs to check on the crew to see how they are doing. An unscheduled rest period should not happen unless the crew calls fatigued.