If by client, you mean turd, then you have to kinda clench your cheeks just a bit but not so much as to cut the turd in half on the way out. Sometimes you won't even splash water or make a "kerplunk" sound if you do it really delicately. Good luck.
"While I appreciate all the business we have done in the past, I find that I can no longer provide the (product, service, etc) at the standards to meet your needs. I'd be happy to refer you to another provider for future business if you would like. Again, thank you for your business in the past." Firm, yet cordial.
Tell him Hillary Clinton wants to have his love child, and you're gonna tell the press about it. That should do the trick.
You should have titled the thread "How do you dump a difficult, gay, Republican client...delicately?" Methinks the thread would have been much more entertaining.
I've had to do that once. I just told them we could provide the services they needed any longer and we would help them transition to a new provider.
I don't dump clients. I encourage (and eventually force) them to make the decision. In my business, "dumping" a customer can come back and bite you later. The 1st thing I do is stop contacting them. The 2nd thing is I stop going out of my way to help them. 3rd, I will not apologize for any mistakes or failures on the part of my company. If the client implies he will go elsewhere, I'll tell him that's his decision. I've come very close to outright telling someone to take their business elsewhere. If the right circumstances presented themselves, that's what you have to do. But if there is a way for the relationship dwindle over time, I'd go that way instead of the direct route. For me, my comfort zone is being direct and confrontational, but in business you have to deal with clients at their comfort level.