Medium Large Transport; Low Wing; 2 Turbojet Eng
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human-Machine Interface; Situational Awareness
Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Weather / Turbulence
Air Traffic Control Issued Advisory / Alert; Air Traffic Control Provided Assistance; Flight Crew Became Reoriented; Flight Crew Requested ATC Assistance / Clarification
Human Factors
Our morning started great. Normal stuff with weather in New York. R1 for different routing and fuel burns. Had an alternate but the forecast wasn't bad enough that we'd actually need to use it. But we had it and enough fuel to handle whatever NY threw at us enroute. All good. Shortly after takeoff; we got a reroute and our first hold. ACARS'd dispatch and talked with the FAs (Flight Attendants) and pax. Calculated bingo fuel. Held for a while; but still good. Cleared out of the hold before our EFC on a new longer route and arrival into LGA. Still had a lot of flying to do and had our speed back to conserve fuel. Many controllers; direct to's; headings assigned; altitudes and speeds later; as we were near CYPER and on the HAARP3 arrival; as assigned; we were handed off and given a descent to 17;000; direct to VALRE and the altimeter. [FO (First Officer)] didn't quite catch the fix; so read back the clearance; asked the controller to repeat the fix and then read back the clearance; including direct to VALRE. We confirmed the 17;000 descent and confirmed and executed the direct to VALRE in the FMS. We had had many plan changes on this now long flight and didn't think in any way that this was at all odd. After we made the turn; the controller barked at us we were off course; we were not cleared to VALRE and to turn back to TRESA and rejoin the arrival. Shaken; we turned back to rejoin; were given a frequency change and were eventually given another hold at VALRE; which we had to depart at new bingo fuel for our alternate; ZZZ. Long flight. Landed safely; albeit late and at the 'wrong' airport for the pax.Unforecasted visibility dropping to 1/2 mile at LGA with rain; thunderstorms and low visibility clearly increased everyone's workloads as we and many other aircraft were enroute with one of the New York Airports as our destination. It was a very fluid and dynamic morning and I thought everyone was doing an awesome job. This one controller who gave us the descent from 190 to 17;000 around CYPER was very busy. I tried to find what exactly was said on liveATC last night; but couldn't locate us; but we heard 'go direct to VALRE'; FO read back 'direct to VALRE' after reading back the rest of the clearance and reconfirming the name of the fix. For the life of me; I don't understand what the reference to VALRE was if not go direct or expect direct or holding. We wouldn't turn off of the arrival unless we were told to and VALRE is 3 fixes further down; anyway. And FO did read back 'direct to VALRE'. If we misheard or misunderstood; that would have been a great time to correct us. Felt pretty bad about increasing his workload and stress though and was upset that I couldn't understand how we had done something wrong.
Air Carrier Captain reported misunderstanding an ATC clearance and deviating from it.