Random get arounds if you don't want to apply for another visa....and since Zhuhai is in the Guangdong Pearl River Delta region
If you ‘forgot’ your visa, or are in a pinch there is also a 5 day “visa on arrival” 落地签 via walking across the Hong Kong border (I remember when I looked a few months ago) in Shenzhen at Futian border where you can exit HK, go upstairs and pay like 1000 RMB and get a 5 day visa. Not 100% sure, but they might exclude Americans from this policy. Technically you are supposed to only stay in Shenzhen because its a Special Economic Zone but maybe you can do a SEZ tour of Zhuhai, Swatow and Xiamen too haha . Just remember to leave on day 5.
Also, I’m pretty sure if you join a tour group from Hong Kong going to Guangdong province, they can spare you a visa in lieu of you shopping and touring with the group (should be about a week long tour) at designated heritage sites like 丹霞地貌,开平碉楼,一个指定的华人之乡.
There's another really really random loophole. University of Macau has a new campus technically on Chinese territory on Hengqin in Zhuhai. So you are in Macau, but also in Zhuhai (Chinese city) at the same time but can only be on the grounds of University of Macau. It's weird, there's like an underwater tunnel, separate communication phone landlines, smugglers, fence-hoppers and illegal immigrants. Anyways, at least you can say you made it there and back.
http://www.universityworldnews.com/article.php?story=20130912115427234