B737 First Officer reported cabin pressure control problems led to a descent to keep cabin altitude from exceeding limits. The aircraft was operating with a MEL already applied. The flight crew continued to destination airport at the lower altitude.

Narrative

Aircraft X pushed from the gate at ZZZ and prior to engine start encountered a left Wing-body overheat indication. Dispatch and Maintenance Control consulted and the left pack deferred under MEL. Aircraft Bleeds and packs configured in accordance with MEL. Release 2 with deferral signed. Departure to the south; climb to initial altitude. Re cleared to 13;000 feet en-route to ZZZ1. Passing 10;000 feet cabin altitude and pressurization checked in accordance with SOP; cabin altitude 2;000 feet and differential pressure indicating that the aircraft was pressurizing. Contacted center and cleared to final cruise altitude of FL230. Passing 17;000 feet rechecked cabin altitude/differential pressure; cabin altitude 6;000 feet and differential pressure same as it was on the 10;000 check. Asked for final cruise altitude of FL190; granted. At level off; cabin altitude 8;000 feet and differential same as at 10;000 feet; check. Requested descent to 10;000 feet and cleared to 13;000 feet. In descent to 13;000 feet cabin altitude warning horn/light sounded and illuminated. Immediate action items complied with and descent to 10;000 requested again and granted. Checklist completed and warnings extinguished. Passenger masks did not deploy. [Requested priority handling]; flight attendants briefed. Talked to passengers. Rest of arrival [was] uneventful; landed and taxied to gate. Write up completed; Chief Pilot debriefed; Dispatch debriefed; ZZZ1 maintenance debriefed face to face in Operations Control.