York Teaching Hospital NHS Foundation Trust, which runs hospitals in York and Scarborough, was forced to cancel some patient appointments over the weekend and today (Monday).
The cyber attacks, using malicious software to hold workers hostage by freezing their computers, started on Friday and first wreaked havoc on some 47 NHS organisations before spreading to more than 100 countries across the world.
In Yorkshire, bone operations and musculoskeletal appointments at hospitals in Scarborough, York and Selby were cancelled this weekend and some patients will still not be seen today after the trust's IT systems were affected by the WannaCry malware.
Hull and East Yorkshire Hospitals NHS Trust were also targeted on Friday but a spokesman said its IT department managed to prevent any significant impact.
A statement by the York NHS trust said appointments today have been cancelled for X-ray bone scans at Scarborough hospital, as well as all outpatient appointments at Selby Hospital except for blood-taking and musculoskeletal physiotherapy.
At Clifton Chapel in York, extended musculoskeletal physiotherapy services are also cancelled today.
The statement said: "All outpatient clinics at York Hospital, Malton Hospital, Bridlington Hospital are going ahead.
"Planned operations are also going ahead as scheduled.
"The situation will be reviewed daily and information will be shared regarding any cancellations to appointments and services later in the week."
The trust was forced to cancel operations of the weekend after the cyber attacks on Friday, which led to its radiology and pathology systems failing, and patients were urged to ensure it was an emergency before attending their A&E departments.
GP practices and Clinical Commissioning Groups (CCGs) across the UK, including some in Yorkshire, were also targeted and some GPs reportedly had no or limited access to patient histories during appointments.
NHS Digital, which is co-ordinating responses and guidance to the malware attack, said it was continuing to work "around the clock" with the National Cyber Security Centre to support NHS trusts affected by the cyber strike.
A spokesperson said: "In addition to providing 24/7 specialist support, we have issued a number of targeted bulletins to further support, guide and reassure NHS staff with cyber-security responsibilities in their organisations.
"NHS Digital is continuing to expand the range of data security services it offers in support of NHS organisations; helping them to take appropriate cyber security measures and respond effectively and safely to cyber security threats."
While many trusts across Yorkshire were unaffected, some were forced to suspend internet access to staff and block incoming external emails until software patches, which update a computers' immunity' were deployed.
Leeds and York Partnership NHS Foundation Trust and the Hull and East Yorkshire trust are continuing to suspend internet access to staff today.
It comes as NHS Digital said that health trusts across England were sent details of an IT security patch that would have protected them from the crippling ransomware attack.
The health service has been rebuked over the weekend for using the outdated Windows XP operating system to store digital information, despite security updates for the software having been discontinued by Microsoft.
But NHS Digital said it had made health trusts aware last month of IT protection that could have prevented the attack.
It said in a statement: "NHS Digital issued a targeted update on a secure portal accessible to NHS staff on April 25, and then via a bulletin to more than 10,000 security and IT professionals on April 27 to alert them to this specific issue.
"These alerts included a patch to protect their systems. This guidance was also reissued on Friday following emergence of this issue."
The Prime Minister's spokesman said today that less than five per cent of NHS systems are still running Windows XP and that other Windows operating systems were also affected.
He said it was "too early to say" what the overall cost of the attack to public coffers would be..
In a statement on behalf of several Yorkshire CCGs, NHS England said the majority of GP practices are as usual today, but that some are "still bringing their IT and clinical systems back online" following the attack.
Patients have been warned that some practices may not yet have full access to records, prescriptions, appointments or telephone systems.
The statement said: "However, all GP practices will be using well-tested contingency plans to ensure that services can
continue to be provided.
"The NHS is asking patients to continue to use the NHS wisely and remember that they can seek help and advice from a range of other sources, such as pharmacies or 111."
People are being advised to attend pre-booked appointments as usual unless contacted directly.