A report by the CQC also found Hull Royal Infirmary (HRI) and Castle Hill Hospital in Cottingham lacked the capacity to meet rising admissions.
It said both hospitals faced "significant challenges" as a result.
But Hull and East Yorkshire Hospitals NHS Trust was praised for its end of life, critical and maternity care.
Both hospitals were rated as "requiring improvement" following the inspection in February.
Prof Sir Mike Richards, chief inspector of hospitals for the CQC, said despite staff efforts to meet patients' needs, shortages "have been felt across a range of services".
"It is of particular concern that the shortage of junior doctors was impacting on the care patients received, leading to delays in assessment and treatment," he said.
The CQC also criticised HRI for being unable to provide a dedicated 24-hour service in its children's accident and emergency department.
In total, the health watchdog identified 20 areas where the trust must improve, including ensuring there are enough suitably qualified and experienced staff on duty, particularly at night and weekends.
It also said the trust must investigate why some staff felt they were bullied into working extra hours.
Ray Grayson, from Unison, said bullying was a "tremendous problem" that required a "culture change" to ensure staff stayed at the hospital.
However, he said: "Staff shortages are occurring right across the NHS. The NHS is not a good place to be at the minute."
Prof Ian Philp, chief medical officer for the trust, told BBC Radio Humberside: "Most of the areas the inspectors identified we had reported to them as our areas of concern and we have got plans in place to address all of those."
The CQC report comes just weeks after Phil Morley quit his role as chief executive of the trust, citing personal reasons for his decision.
John Saxby, the trust's interim chief executive, said investment in nursing staff had recently risen from Â£450,000 to Â£1m.
He vowed to investigate and, if found, "eliminate" bullying.
A shortage of doctors and nurses is affecting the care of patients at two East Yorkshire hospitals, the Care Quality Commission (CQC) has said.