CRJ-200 Captain reported a cabin pressure control failure after departure. With one pack on MEL and the failure of the second pack; the flight crew elected to perform a return to the departure airport.

Narrative

We were to operate a Company flight from ZZZ to ZZZ1; FO (First Officer) was PF (Pilot Flying). Aircraft X had the right hand pack on MEL. Before departure; we reviewed the procedures for operating single pack; including procedures to follow if the working pack failed. We departed ZZZ and climbed to FL230 when we received status messages for L Pack OFF and L 10TH ARM OPEN. The only procedure that was appropriate to run at the time was the Unpressurized Flight Procedure (Packs Off) QRH; as stated by the MEL. I had informed ATC we were experiencing a pressurization issue and they declared priority handling for us. I asked for a lower altitude before continuing with the QRH. ATC kept asking us questions; which was repeatedly delaying the QRH. We started a turn back to ZZZ and a descent to 10;000 ft. while I continued with the QRH. Before reaching the last item on the QRH; we received the Cabin Altitude warning message. I stopped the current QRH and switched to the QRH for this message. At the time; our cabin altitude was 10;800 ft. and already descending rapidly; so I elected not to perform the actions for donning crew masks for deploying passenger oxygen masks. Less than a minute later; we leveled at 10;000 ft. MSL with a cabin altitude of 8;500 ft. with the cabin altitude control in manual mode. I then talked to the FA (Flight Attendant) and informed her of the situation and she had no concerns to inform me of. Returning to ZZZ; we were 1;500 lbs overweight for a normal landing; so we did a holding pattern north of the airport for approx 15 minutes. This is when I completed the Unpressurized Flight Procedure QRH. We performed a normal landing to Runway XX. I stopped the aircraft upon exiting; where fire trucks followed us to the gate. I observed all passengers deplaning; they appeared sweaty (it had reached 32 Degrees Celsius in the cabin) and a few passengers were complaining about their ears being uncomfortable; but nothing worse. Almost all of them reboarded after the aircraft swap so they must not have been too scared.Dispatching a plane that is single pack in general really is not a great idea. Having a more detailed procedure for pack failures in the MEL might be helpful too. I'm told it was likely a bad connection to the L 10th stage valve that starved the left pack; If I had tried opening the 10th stage ISOL and restarted the pack with air from the right hand engine; it likely would have fixed the problem; but I had not considered it at the time. I might also suggest maybe a maintenance inspection of the only working pack when dispatching flights further; especially when departing maintenance stations. Maybe it should be a 3 day MEL instead of 10. Also; ATC could have given me more time. I got 3 frequency changes and asked souls on board and fuel remaining before finishing a 7 line QRH. However; if I had ignored them and finished my QRH; I might have avoided the Cabin Altitude warning.