It may. Or may not. In general the golden rule is to not provide more info than asked but also not to withhold any information or even appear to do so. You might consider getting a new passport but to a visa-officer that typically raises more questions than it gets out of the way, so I would avoid that. Also note that there is an annual quota of 1000 for UK citizens so it might be just bad luck should you be refused. That being said, good luck! Do give us an update when you reveive an answer!
Older version of my answer, now outdated given OP edit:
STOP! Stop what you are doing right now and get your own research going. You might be getting SCAMMED. Do not rely on any company or advisor.
First, I do not know of a "working holiday visa" for going to China. It is not listed on the list of Chinese visa types on Wikipedia. There is exactly one website which results high on Google talking about such a thing and besides the horrible make of the website, they seem absolutely unprofessional in not making a distinction between China and Taiwan. In fact there is such a program for Taiwan, not China. (Programs for China are only considerations for now.) There also exists one for Hong Kong but then mind you will be working in Hong Kong only, not in China, which you should see as a separate country for the purposes here.
Given the story of how you got that ugly stamp in your passport, you should be a lot more careful in terms of what companies advice you trust.