The Care Quality Commission (CQC) raised concerns over numbers of doctors and nurses at South Tees Hospitals NHS Foundation Trust.
It warned it must ensure it has "enough qualified and experienced staff", particularly in A&E and surgical wards.
But, the trust said the report had "many positives" and staffing levels had improved.
The trust, which is responsible for James Cook University Hospital, in Middlesbrough, The Friarage Hospital, in Northallerton, North Yorkshire, and a range of community services, was inspected in December.
The hospitals were rated as "requires improvement" both in end-of-life care and overall, while other core services were rated "good".
While finding the trust "provided services that were caring, responsive and well led", the CQC concluded standards needed to be raised in areas including staff training and the upkeep of patients' records.
Professor Sir Mike Richards, chief inspector of hospitals, said: "We came across numerous examples of excellent care and patients and their relatives were happy with their experience of care and treatment.
"However, I am concerned that shortages of staff in some departments may affect patient care.
"While there have been moves to improve the recruitment process, the trust must continue to make this a priority."
Trust bosses said they "welcomed" the report in which "89 of the 105 individual ratings (84%) were either good or outstanding and a number of areas of excellent and outstanding practice were also identified, particularly in maternity services".
Chief executive Professor Tricia Hart said: "Since the CQC visit, we have already made significant improvements to staffing levels, particularly overnight, and continue to push hard on nurse recruitment, recently employing nurses from Italy, Portugal and Romania, although this does remain a national issue."
Staff shortages could threaten patient care at a north-east England health trust that "requires improvement".