Not sure if I understand you correctly. You say only 1 page of the 20 is detecting plagiarism, but then go on to say that you suspect the other pages are using the ‘same method’ to hide from Copyscape. What method are you referring to?
Assuming it’s just the one page that’s being detected for plagiarism, I would just remove it completely or rewrite it. From my understanding, there’s no such thing as a ‘duplicate content penalty’ – i.e. google isn’t going to punish your site for plagiarism. The flipside to this is that although you won't be punished, your content likely won't rank either. Why? Because google doesn’t like showing duplicate content in SERPs. Because it's already indexed and ranked the original content your writer plagiarised from, it will continue to show that version in the SERP and essentially just ignore your one. From an SEO standpoint, this piece of content won't be that useful to you if you can't rank it.
In short, google isn't going to punish your site so you don’t need to worry about any long-term damage, however, your plagiarised content likely won't rank either so it's essentially useless. The best option would be to either rewrite it or completely remove it.
Also, as a side note, I wouldn't worry about your content writer using existing competitor content to form new content for you. As long as it's not being detected as plagiarism and it's not a complete rip-off (i.e. talks about the exact same topics in the exact same order, using the exact same language and exact same analogies and examples etc.), then it's fine. Most content writers (at least with the ones I've dealt with anyway) just rewrite existing competitor content in their own words. Unless you're paying them a moderate amount which would realistically cover their time for both research + writing, it's fair to assume that most are going to skip the 'research' part and just focus on the 'writing' aspect by finding content they can rewrite.
