Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Communication Breakdown
Deviation - Altitude Crossing Restriction Not Met; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Provided Assistance; Flight Crew Executed Go Around / Missed Approach
Human Factors; Procedure
On DATE while proceeding from ZZZ to ZZZ1 on a ferry flight with an employee's daughter onboard; we were diverted to ZZZ2 for a passenger recovery flight. We were proceeding direct to ZZZZZ on the RNAV GPS Approach. We were instructed to hold at ZZZZZ as published at an altitude of 6;800 ft. due to an aircraft that would be crossing our missed approach path until ATC made radio contact with the traffic. Approximately 5 NM prior to ZZZZZ; we were cleared for the approach. I proceeded to set 3;700 ft.; the FAF Altitude; in the ALT window; armed the VPATH and the APP mode. The aircraft proceeded to descend to cross ZZZZZ at or above 5;400 ft. A few minutes later; just prior to MREEE; at approximately 6;000 ft. we received a LOW ALTITUDE ALERT from ATC. He requested that we climb back up to 6;800 ft. We had the field in site and were established on the approach so; we cancelled our IFR flight plan and we were then cleared for the visual approach. We did not receive any TERRAIN WARNINGS; there was no yellow displayed on our PFD/MFD in regards to terrain. At the time; the SIC was on the CTAF frequency announcing our position and commencement of the RNAV GPS approach. Upon further query; ATC stated that they had asked us to maintain 6;800 ft. until ZZZZZ before commencing the approach. I did not hear that altitude restriction and only read back the cleared for the approach authorization. I'm not sure why we had that restriction; as we were in VFR conditions. All the high terrain was to our southeast; there was no other traffic in the area or displayed on our TCAS and 5;400 ft. was the crossing altitude for MREEE on the approach. The safety of the crew; passenger and aircraft was never in question and we had situational awareness throughout the approach.
Captain reported missing an ATC communication; resulting in Low Alt alert and a Missed approach and go around.