The Care Quality Commission's (CQC) assessment followed inspections in May.
Inspectors had received information of concern about these two services at the Wirral hospital.
Inspectors highlighted some areas for improvement such as patient flow and bed occupancy with pressure on hospital services recognised as an issue nationally.
The on-going work to enhance patient flow, including the provision of Same Day Emergency Care services was also recognised.
The rating for medical care dropped from 'good' to 'requires improvement' whilst urgent and emergency care services remain at 'requires improvement'.
The hospital was rated 'good' in areas of being 'caring, effective and well-led' but improvement is needed in the areas of patient safety and being responsive.
In the 'effective' domain, inspectors noted staff worked together as a team to benefit patients and provide safe care and followed policies and guidelines to plan and deliver care.
In the 'Caring' domain, inspectors highlighted that staff listened to and understood people's needs, views and wishes, responding to people's needs in the moment and acted to minimise any discomfort, concern or distress.
The caring nature of staff supporting other staff members' strengths, abilities, aspirations, culture and protected characteristics was also highlighted by inspectors.
Also, recognised was the service caring about and promoting the wellbeing of staff and supporting them to always deliver person-centred care.
In the 'well-led' domain, inspectors found staff and leaders had a shared a commitment to improving services for patients and their families. The service 'demonstrated a good standard of care, had a shared direction and culture with inclusive leaders at all levels who embodied the culture and values of the organisation'. The inspectors found leaders led effectively with integrity, openness and honesty.
During the assessment, inspectors also observed inconsistencies in 'oversight of patients in the emergency department and in transfer to specialities'. Referring to one incident, the report read: "In the Clinical Decision Unit (CDU) one patient was left 19 hours before medical review due to an administrative error which meant the patient was left under the ED consultant name overnight and not transferred."
A statement in the CQC report read: "During our inspection we identified areas of concern which required the trust to take immediate action. These were escalated to the trust's senior leaders who acted to make improvements.
"The service did not always provide and maintain safe systems of care, in which safety was managed, monitored and assured.
"Most processes and policies to plan and deliver people's care and treatment were in line with legislation and current evidence-based good practice and standards. However, people were not always cared for in the right place."
During the visit, CQC inspectors looked at 20 sets of patient clinical records, spoke with 10 patients, five family members and 43 members of staff.
The inspectors found that: "Leaders had implemented an action plan to decrease the high number of inpatient falls and hospital acquired pressure ulcers.
"The service didn't always have enough qualified, skilled and experienced staff due to shortfalls in mandatory training in some staff groups.
"Senior leaders acknowledged that patient flow was one of the biggest challenges within the hospital with bed occupancy being regularly above 95%".
Staff were described by people as 'kind and caring, whilst working under significant pressure'.
Staff knew how to identify adults and children at risk of, or suffering, significant harm and worked with other agencies to protect them.
In its assessment of urgent and emergency services at Arrowe Park, inspectors found: "People were frequently treated in unsuitable and unsafe areas, including corridors, with little privacy or dignity.
"Leaders had improved staffing levels; however, numbers didn't always meet demand within the department which could place people at risk".
In terms of patient safety, the report stated: "The service did not always make sure there were enough qualified, skilled and experienced staff due to shortfalls in mandatory training for medical staff and vacancies for therapy staff.
"The service did not consistently ensure medicines and treatments were safe and met people's needs, capacities and preferences. However, staff recognised and reported incidents and near misses.
"Managers investigated incidents and shared lessons learned with the whole team and the wider service. Staff understood how to protect patients from abuse and the service worked well with other agencies to do so."
The safeguarding team was described as 'approachable' with 'staff familiar with the process and who the designated safeguarding nurse is if any incidents occurred'.
The service provided care bags containing sensory aids (eye mask, headphones, and fidget tools) for autistic people and people with disabilities.
Janelle Holmes, chief executive at Wirral University Teaching Hospital said: "We acknowledge and welcome the report by the Care Quality Commission.
"This independent feedback helps us continue to make improvements to our services as we continue to navigate what is a challenging time for all NHS services.
"It is reassuring to see that inspectors recognised aspects of our services as 'Good' and highlighted how effective our services are. They also acknowledged how kind and caring, our staff are while working under significant pressure.
"As with other Trusts, our Urgent and Emergency Care Services have faced very high levels of attendances throughout the year.
"We are close to completing our brand new Urgent and Emergency Care Facility which will greatly improve patient experience when it opens next year.
"Our staff continue to work extremely hard to provide quality care for our patients and we have plans in place to ensure our staff feel supported as we head into a busy winter period."