Wirral University Teaching Hospital Trust said it was still in the process of recovering some services.
Bosses said the main clinical system was being restored and the incident had now been downgraded to a “business continuity incident”.
“Some services will continue to be affected this week as systems are restored,” a hospital spokesperson said.
“Emergency treatment is being prioritised but there are still likely to be longer than usual waiting times in our emergency department and assessment areas,” they added.
The hospital has asked anyone with an outpatients appointment to come to their appointment as planned.
They repeated calls for people to only attend the emergency department in genuine emergencies.
Those with non-urgent health concerns were asked to use NHS 111, walk-in centres, urgent treatment centres, GPs or pharmacies.
A Merseyside Police spokesperson said they were working with the National Crime Agency and National Cyber Security Centre “to provide the necessary support and assistance”.