Citationjet pilot reported they descended below the assigned altitude crossing restriction for an approach to a non towered airport and received a terrain warning. A go around was completed and a second successful approach to landing was made.

Narrative

Flight was ZZZ1-ZZZ. I was cleared for the RNAV X approach into ZZZ; straight in from ZZZZZ. ATC instructed me 'cleared direct ZZZZZ; direct ZZZ airport; maintain at or above 7000 ft. until established; cleared for the RNAV X into ZZZ'. What I apparently heard was 'maintain at or above 5000 ft.' and I believe that this is what I read back to ATC. I continued my descent to about 5100 ft.; which I arrived at around ZZZZZ1 waypoint; the FAF; until the system activated; at which point I executed the proper procedure for such a warning: disconnecting autopilot and initiating an immediate max power climb. Because of the confusion of altitudes; I did not execute the published missed approach procedure. Instead; I ended up climbing towards the airport to between 7000 ft. and 8000 ft. before switching from CTAF back to ZZZ Center and being assigned an altitude of 8000 ft. by ATC. I told ATC that I had gone missed and that I had received a terrain warning after being told to maintain 'at or above 5000 ft. until established' ATC seemed confused by this. Upon going back and listening to the audio playback; ATC had actually instructed me to maintain 'at or above 7000 ft.'; NOT the 5000 ft. altitude that I thought I heard. Upon review of the flight track; it appears that I missed terrain by about 500 ft. by descending to about 5100 ft. This occurred late in the afternoon after a long day of continuous flight plan changes due to deteriorating weather in the midwest. Though I had a good night's rest the night before; I believe mental fatigue was a major factor in what occurred because I had spent so much energy throughout the day trying to come up with fuel stops and legal/safe alternates that would allow me to safely bring passengers to their final destination of ZZZ2. I had changed routing many; many times during the day; and weather updates had continually required me to make additional amendments to my routing over the course of the day. The reason that an ATC assignment to maintain at or above 5000 ft. until glide path intercept did not seem too outrageous to me was because the airport sits at about 4000 ft. and the airport is; as far as I knew; not in a mountainous area. I had never been to ZZZ; it didn't occur to me that there would be any worrisome terrain. Contributing to this error was the fact that; assuming I had indeed read back '5000 ft.;' ATC did not seem to catch my erroneous read back. Flying single-pilot meant that there was no second set of ears in the cockpit to catch my mistake. I ended up shooting the approach again and making a normal landing on the published approach. I am currently planning on leaving this job at the end of the year and pursuing flight in 2-pilot operations so as to increase safety margins; among other things.