Is money earned from teaching someone without having a teaching certificate permissible or forbidden, given that the student benefited, and is it permissible to continue teaching?

There is no harm in taking a fee for teaching a man's sons. However, you must repent for lying to the woman by claiming to be a teacher. Nevertheless, there is no harm in taking the fee because you performed the work. There is no objection to continuing to teach if you are qualified, provided that you do not claim to be a teacher.