Date: 202107
Local time of day: 1201-1800
Location: ZZZ.TRACON, US
Aircraft: Medium Transport; Low Wing; 2 Turbojet Eng
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Ferry / Re-Positioning
Flight phase: Descent
Airspace: Class C ZZZ
ATC / advisory: TRACON ZZZ
Reporter organization: Government
Reporter function: Approach
Reporter qualification: Air Traffic Control Fully Certified
Human factors: Situational Awareness; Communication Breakdown
Second reporter function: Pilot Not Flying; First Officer
Anomaly: ATC Issue All Types; Aircraft Equipment Problem Less Severe; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT; Inflight Event / Encounter Fuel Issue
Detector: Person Air Traffic Control
When detected: In-flight
Result: Air Traffic Control Issued New Clearance; Air Traffic Control Issued Advisory / Alert; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Returned To Clearance
Contributing factors: Aircraft; Airspace Structure; Human Factors; Procedure
Primary problem: Human Factors

Narrative: Aircraft X was issued a descent to 2500 feet but as soon as I said 2000; the pilot started to read back 2000. This caused the 500 (of the 2500) to be blocked and when I un-keyed; I just heard the call sign portion of the transmission. The aircraft descended to 2000 feet and asked for an altimeter setting and I noticed the aircraft below the 2500 minimum vectoring altitude (MVA) and assigned altitude. I issued low altitude alert and instructed them to climb to 2500 feet.  I missed the read back. This all stemmed from the aircraft departing ZZZ with a known mechanical issue and low on fuel. They had diverted to ZZZ earlier (unknown to me) and was trying to reposition back to ZZZ1.They departed ZZZ1 minimum fuel and would not accept an approach at ZZZ1 that was consistent with the flow of traffic. I should have [requested priority handling] for the pilot immediately and handled it as one. The pilot put the FAA in a tough situation as I was trying to do my best to help the pilots who were in a bind; but I also missed many things I should have done.I did not ensure the aircraft had the weather and NOTAMs at ZZZ1; [request priority handling] for the aircraft; advise them of their position when vectoring them on to the localizer; allowed them to intercept the localizer inside the final approach fix. I did not catch the lack of read back when I issued 2500 MVA altitude and they over keyed me with a read back of 2000. Issue a brasher to the pilot.  I need to make sure all the phraseology and regulations are followed on our side (ATC)However; Aircraft X's principle operations inspector needs to investigate if the flight from ZZZ to ZZZ1 was conducted legally and safely. It was very apparent the urgency to land immediately after taking off from ZZZ. This should not have happened. This caused me to make shortcuts and put the FAA at risk of liability as well as put the pilots at risk of missing something like landing on runway XX at ZZZ1; a closed runway. The pilots were wanting to land on runway XX at one point.


Second narrative: Upon departure from ZZZ to ZZZ1; a short 5 minute or so flight; we were given a heading and an initial altitude of 3000 feet on initial departure. Given we were taking the plane in for maintenance; the plane had enough fuel to meet the minimum FAR requirements for an IFR flight. While setting up for the approach at hand; we were told by the controlling facility to descend and maintain 2;000 feet which was read back. A couple minutes went by as we were being vectored in for the approach and ATC gave us a low altitude alert followed by a command to 'descend and maintain 2;500 feet MSL'. Having already been cleared down to 2000 feet initially after we asked for lower; we read back the instruction for 2;500 feet followed by a 'we were initially cleared down to 2;000 feet MSL'.At that point; the controlling facility told us that they weren't going to debate over it and we instigated the climb to 2;500 feet. We only meant to reiterate the initial 2;000 foot altitude instruction for clarification purposes; not for argumentative purposes; as that was the initial instruction given to us after all. After landing; we received a phone call to contact the appropriate controlling facility to clarify the situation. We pulled the tapes from the appropriate controlling facility at the time of the incident and the tapes coincide with our initial hearing of '2;000 feet MSL;' not '2;500 feet MSL'. The altitude is clearly heard as 2;000 feet being the instruction given and 2000 feet being the instruction read back; not '2;500'. In order to keep a situation like this from reoccurring; an articulate tone should be adopted in order to hear and say the correct altitude assigned. Given that a weather system was rolling through as well; that added extra workloads on both us; the pilots; along with the controlling facility.


Synopsis: TRACON Controller and air carrier pilot reported the pilot descended below the Minimum Vectoring Altitude due to a misunderstanding of the assigned altitude between the two parties.