The phrase "serious offence" refers to a very serious and significant wrongdoing or crime, which is typically punished severely under the law. Full definition
She was found guilty of the more serious offence of murder. (politics.co.uk)
How will they ensure that prisoners who are guilty of serious offences but serving short sentences are not given the vote? (conservativehome.blogs.com)
Don't do so for serious offences with long incarceration periods because it will automatically create employment gaps, which will raise some question. (prime-resume.com)