What are the chances of getting a problem in the PR process in Canada?
Canada uses your name as it appears in your passport. It creates some administrative issues as most computer systems used require a first and last name but is not a legal impediment. Once in Canada you will run into problems everywhere, from banking to school to healthcare to driving. Virtually none of those services have the capacity to accept only one name, regardless what your PR document says. A new passport with a first and last name will make your life considerably easier.