In some countries, the mother will have full legal custody by default if the father is not mentioned on the birth cert. There would be no need to go to court to get full legal custody. I've been through this situation and didn't need a court order. However if the father is registered on birth cert then yes a court order is required.As scylla has said, you need to obtain full legal custody from the court. The guardianship is not enough.
There is no "single parent" visa. You need to qualify for immigration like everyone else. https://www.canada.ca/en/services/immigration-citizenship.html